111 results on '"M.A. Hasan"'
Search Results
2. Contributors
- Author
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Manar E. Abdel-Raouf, Akil Ahmad, Sajid R. Ahmad, Nisar Ali, Mohammed B. Alshammari, Muzammil Anjum, Ph.D, Shahid P. Ansari, Tadele Assefa Aragaw, Chaudhry Arslan, Adane Adugna Ayalew, Muhammad Ayub, Muhammad Bilal, Mohsin R. Dustgeer, Xiaoyan Gao, Syed H. Gillani, Suresh Ghotekar, Chirag Batukbhai Godiya, Md. Morshedul Haque, Abdulraheim M.A. Hasan, Mohamad Nasir Mohamad Ibrahim, Mustapha Omenesa Idris, Ahmad F. Ismail, Juhana Jaafar, Mohammad Jawaid, Asim Jilani, Roziana Kamaludin, Ansa Kanwal, Azeem Khalid, Ph.D, Adnan Khan, Asma Khatoon, Mohsin Khurshid, Lau Kia Kian, Wei Liu, Ph.D, Sumeet Malik, Muhammad A. Mirza, Aleid G. Mohammad S, Abdul Nasir, Lingli Ni, Nahin Mostofa Niloy, Mohd Hafiz Dzarfan Othman, Fahmida Parvin, Samia Qadeer, Ph.D, Mukhlis A. Rahman, Haroon Rashid, Arpita Roy, Luis A.M. Ruotolo, Naheed Saba, Muhammad Sarmad, Siti H.M. Setapar, Muhammad B. Shakoor, Affia Siddique, Jamal A. Siddique, Shafi M. Tareq, Waheed Tariq, Nafeesa Tayyab, Yong Yang, Asim Ali Yaqoob, Hazirah S. Zakria, and Muhammad Zubair
- Published
- 2023
3. Neural mechanisms of emotional health in traumatic brain injury patients undergoing rTMS treatment
- Author
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T. Sultana, M.A. Hasan, X Kang, V. Liou-Johnson, M.M. Adamson, and A. Razi
- Abstract
Emotional dysregulation such as that seen in depression, are a long-term consequence of mild traumatic brain injury (TBI), that can be improved by using neuromodulation treatments such as repetitive transcranial magnetic stimulation (rTMS). Previous studies provide insights into the changes in functional connectivity related to general emotional health after the application of rTMS procedures in patients with TBI. However, these studies provide little understanding of the underlying neuronal mechanisms that drive the improvement of the emotional health in these patients.The current study focuses on inferring the effective (causal) connectivity changes and their association with emotional health, after rTMS treatment of cognitive problems in TBI patients (N=32). Specifically, we used resting state functional magnetic resonance imaging together with spectral dynamic causal model to investigate changes in brain effective connectivity, before and after the application of high frequency (10 Hz) rTMS over left dorsolateral prefrontal cortex. We investigated the effective connectivity of the cortico-limbic network comprised of 11 regions of interest which are part of the default mode, salience and executive control networks, known to be implicated in emotional processing.The results indicate that overall, among extrinsic connections, the strength of excitatory connections decreased while that of inhibitory connections increased indicating reduction in the activity of all the inspected brain regions after the neuromodulation. The cardinal region in the analysis was dorsal anterior cingulate cortex which is considered to be the most influenced during emotional health disorders.Our findings implicate the altered connectivity of dorsal anterior cingulate cortex with left anterior insula and medial prefrontal cortex, after the application of rTMS, as a potential neural mechanism underlying improvement of emotional health. Our investigation highlights the importance of these brain regions as treatment targets in emotional processing in TBI.
- Published
- 2022
4. Green Polymers and Their Uses in Petroleum Industry, Current State and Future Perspectives
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Mohamed Keshawy, Manar El-Sayed Abdel-Raouf, and Abdulraheim M.A. Hasan
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Engineering ,Petroleum industry ,business.industry ,Biochemical engineering ,State (computer science) ,Current (fluid) ,business - Abstract
The concept of green chemistry has been established to find safe methodologies and environmentally benign solutions for the present and the onset problems. In this regard, extensive work has been carried out worldwide to replace the currently used materials with green ones. The terminology green relies on all the non-pollutive or the degradable materials regardless of their source. Therefore, there are biobased green materials and non-biobased green materials. This review sheds light on several green polymers used in different petroleum industries. The polymers are reviewed according to the stage of oil processing in which they are applied. Furthermore, different modification methodologies of natural polymers are revised. Also, the role of green non-biopolymers in different petroleum industries is investigated. It is worth mentioning that we concentrate our efforts on the utilization of different natural polymers in petroleum applications. Thereafter, some natural polymers such as chitosan and cellulose and their derivatives were specifically reviewed.
- Published
- 2022
5. Analysis of the response of damped and parametrically driven, strongly anharmonic Klein-Gordon chain - Part 1: Phase locked traveling breathers
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M. Gzal, V. Kislovsky, M.A. Hasan, and Y. Starosvetsky
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General Mathematics ,Applied Mathematics ,General Physics and Astronomy ,Statistical and Nonlinear Physics - Published
- 2023
6. Serum Unbound Bilirubin Versus Transcutaneous Bilirubin in Jaundiced Term Neonates
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A.E. Elsadek, M.A. Elbishry, M.A. Hasan, and E.G. Behiry
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medicine.medical_specialty ,Transcutaneous bilirubin ,business.industry ,Bilirubin ,General Medicine ,Jaundice ,Term neonates ,Gastroenterology ,Serum bilirubin ,chemistry.chemical_compound ,chemistry ,Transcutaneous bilirubinometry ,Internal medicine ,Medicine ,medicine.symptom ,business - Abstract
Transcutaneous bilirubin (TcB) consists of the skin-deposited bilirubin. Free bilirubin speaks to the protein-unbound bilirubin (UB) that can go into the tissues. We expected to portray the connection between UB-TcB and study the section of UB into the skin. We tentatively enlisted 40 neonates and we estimated TcB, UB and all out serum bilirubin. Different locales TcB estimation was performed. TcB has a comparative relationship with UB and TSB.There is asignifigant relationship with TSB, DSB(direct serum bilirubin) between embittered term neonates when contrasted with solid term neonates. There is a critical relationship between free bilirubin in embittered term neonates when contrasted with solid ter.mTcB test doesn't appear to straightforwardly supplant UB estimation. Be that as it may, TcB and UB are connected ,UB adds to the skin bilirubin statement yet it isn't the main bilirubin species estimated by transcutaneous bilirubinometry.
- Published
- 2020
7. A preliminary study on liquid crystalline epoxy curatives from natural abietic acid
- Author
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Maher A. El-Sockary, Hussin I. Al-Shafey, Ashraf M. El-Saeed, R.A. El-Ghazawy, and Abdulraheim M.A. Hasan
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Thermogravimetric analysis ,Materials science ,020209 energy ,02 engineering and technology ,Thermotropic crystal ,Catalysis ,chemistry.chemical_compound ,Differential scanning calorimetry ,020401 chemical engineering ,Geochemistry and Petrology ,Liquid crystal ,0202 electrical engineering, electronic engineering, information engineering ,0204 chemical engineering ,Abietic acid ,lcsh:Petroleum refining. Petroleum products ,Curing (chemistry) ,Renewable Energy, Sustainability and the Environment ,Process Chemistry and Technology ,Organic Chemistry ,Epoxy ,Fuel Technology ,Chemical engineering ,chemistry ,lcsh:TP690-692.5 ,visual_art ,visual_art.visual_art_medium ,Glass transition - Abstract
Abietic acid was used for preparing two thermotropic liquid crystalline curing agents with different mesogens (LCC1 and LCC2). The chemical structure of the synthesized LCCs was investigated by FTIR and 1HNMR spectroscopic techniques. Their LC phase transition temperatures and texture were identified by differential scanning calorimeter (DSC) and polarized optical microscope (POM), respectively. Both LCCs show nematic droplet texture -as revealed by POM micrographs- but different isotropization temperatures. Eventually, they were used for producing epoxy coatings using a multifunctional abietic acid based epoxy resin; prepared in our previous study. Cured coats were evaluated by standard coating testing and compared with a commercial curing agent. Dynamic mechanical (DMA) and thermogravimetric (TGA) analyses show that the fully bio-based cured epoxies by either LCCs possess high glass transition temperature (Tg), high modulus (G′) and improved thermal stability. Keywords: Abietic acid, Epoxy, Liquid crystal, Dynamic mechanical analysis, Thermogravimetric analysis
- Published
- 2019
8. Applications of guar gum and its derivatives in petroleum industry: A review
- Author
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Manar El-Sayed Abdel-Raouf and Abdulraheim M.A. Hasan
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02 engineering and technology ,010402 general chemistry ,01 natural sciences ,Catalysis ,chemistry.chemical_compound ,Geochemistry and Petrology ,lcsh:Petroleum refining. Petroleum products ,chemistry.chemical_classification ,Guar gum ,Renewable Energy, Sustainability and the Environment ,business.industry ,Process Chemistry and Technology ,Organic Chemistry ,Natural polymers ,Polymer ,021001 nanoscience & nanotechnology ,Natural resource ,0104 chemical sciences ,Renewable energy ,Fuel Technology ,chemistry ,Petroleum industry ,lcsh:TP690-692.5 ,Petroleum ,Environmental science ,Biochemical engineering ,0210 nano-technology ,business - Abstract
Green chemistry approach is growing rapidly and the replacement of petroleum derived materials with bio-based materials is greatly enhanced for environmental and health concerns. Polymers from renewable natural resources have attracted an increasing amount of attention over the last three decades, predominantly due to two main reasons: firstly environmental concerns, and secondly the realization that our petroleum resources are finite. Natural polymers are available in large quantities from different renewable sources, while synthetic polymers are produced from non-renewable petroleum resources. They include proteins and polysaccharides. Among the most important polysaccharides is Guar gum which has been extensively used in a wide range of applications due to its unique properties. The objective of this review is to focus the light on Guar Gum and its derivatives. The chemical structure, sources, chemical modification, physical properties (mechanical properties and thermal stability), and some potential industrial applications are reviewed and discussed. Keywords: Natural polymers, Guar gum, Chemical modification, Drilling fluids, Thickening agents
- Published
- 2018
9. Atomic force microscopy investigation of smart superabsorbent hydrogels based on carboxymethyl guar gum: Surface topography and swelling properties
- Author
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Abdulraheim M.A. Hasan, Mohamed Keshawy, and Manar El-Sayed Abdel-Raouf
- Subjects
General Materials Science ,Condensed Matter Physics - Published
- 2022
10. Author response for 'Adaptive modulation index‐based controller for suppression of circulating current in parallel inverters of autonomous microgrid'
- Author
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M.A. Hasan and S. K. Parida
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Index (economics) ,Control theory ,Computer science ,Circulating current ,Link adaptation ,Microgrid - Published
- 2020
11. Importance and Potentiality of Underutilized Lakoocha (Artocarpus lakoocha Roxb) Fruit of Tripura
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S. Islam, M.A. Hasan, Tamanna Perween, Sonam Bhutia, and P.S. Munsi
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Traditional medicine ,Artocarpus lakoocha ,Biology - Published
- 2018
12. Effect of Different Dose of NPK on Flower Phenology of Dragon Fruit
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Tamanna Perween and M.A. Hasan
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Horticulture ,Phenology ,Biology - Published
- 2018
13. Skyrme–Hartree–Fock–Bogoliubov Calculations of Even and Odd Neutron-Rich Mg Isotopes
- Author
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A.H. Taqi and M.A. Hasan
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Physics ,Nuclear physics ,Isotope ,Nuclear Theory ,Hartree–Fock method ,General Physics and Astronomy ,Neutron ,Nuclear Experiment - Abstract
Using the Skyrme functional with SIII, SKM*, SLy4, and UNE0 sets of parameters and the Hartree–Fock–Bogoliubov mean-field method; the ground-state properties of even-even and even-odd neutron-rich Mg isotopes have been investigated. The results of calculations of the binding energy per nucleon (B/A), the one- and two-neutron separation energies (Sn and S2n), proton and neutron rms radii, neutron pairing gap, and quadrupole deformation parameter (B2) have been compared with the available experimental data, the results of Hartree–Fock–Bogoliubov calculations based on the D1S Gogny force, and predictions of some nuclear models such as the Finite Range Droplet Model (FRDM) and Relativistic Mean-Field (RMF) model. Our results show good agreements in comparison with the experimental data and the results of the mentioned models.
- Published
- 2021
14. Cellulose-Based Superabsorbent Hydrogels
- Author
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Abdulraheim M.A. Hasan and Manar El-Sayed Abdel-Raouf
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chemistry.chemical_compound ,chemistry ,Chemical engineering ,Self-healing hydrogels ,Cellulose - Published
- 2019
15. Climate Change and Mango Production
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M.A. Hasan and Shuvadeep Halder
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Agroforestry ,Production (economics) ,Environmental science ,Climate change ,General Medicine - Published
- 2020
16. Anatomical and physiological traits of broad bean (Vicia faba L.) seedling affected by salicylic acid and salt stress
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J.H. Hamza, M.A. Hasan, Muwafaq Abdul-Razzaq Al-Naqeeb, M.H.K. AL-Baldawwi, Sudad K. Al-Taweel, and H. A. Al-Amrani
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0106 biological sciences ,0301 basic medicine ,chemistry.chemical_classification ,Control treatment ,biology ,Soil Science ,Salt (chemistry) ,Plant Science ,Horticulture ,Vascular bundle ,biology.organism_classification ,01 natural sciences ,Vicia faba ,03 medical and health sciences ,chemistry.chemical_compound ,030104 developmental biology ,chemistry ,Seedling ,Laboratory experiment ,Agronomy and Crop Science ,Salicylic acid ,010606 plant biology & botany - Abstract
A laboratory experiment was carried out at the College of Agriculture University of Baghdad in 2017. The aim was to improve the anatomical and physiological traits of broad bean seedling under salt stress by soaking it in salicylic acid. The concentrations of salicylic acid were 0, 10, and 20 mg L-1 and the electrical conductivity levels were 0, 3, and 6 dS m-1. The complete randomized design was used with four replications. The increasing of salicylic acid concentration up to 10 mg L-1 led to increasing the stem cortex thickness, stem vascular bundles thickness, and root cortex thickness significantly by (34.9,36.7,and 55 µm) respectively, while the treatment of 20 mg L-1 led to decreasing these traits by (28.2, 27.8, and 48.1 µm), compared to control treatment (33.8, 35.9, and 53.8 µm), respectively, and the interaction of studied factors led to increasing those traits up to 10 mg L-1 and then decreased up to 20 mg L-1 of salicylic acid under each level of electrical conductivity. Therefore, it is recommended to soak the broad been seeds with 10 mg L-1 salicylic acid to improve the anatomical traits of seedlings and increase their tolerance to salt stress up to 6 dS m-1.
- Published
- 2018
17. Liquid Emulsion Membrane Stabitiy Studies for Removal of Nickel from Liquid Aqueous Waste
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M.A. Hasan, A.A. El-Taweel, and S.T. Atwa
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Nickel ,Aqueous solution ,Chromatography ,Membrane ,Chemical engineering ,Chemistry ,Emulsion ,chemistry.chemical_element ,General Chemistry - Published
- 2015
18. Removal of Lead from Liquid Waste Using Liquid Emulsion Membrane
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S.T. Atwa, A.A. El-Taweel, S.A. Shama, E.A. El-Sofany, and M.A. Hasan
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Lead (geology) ,Chromatography ,Membrane ,Chemistry ,Emulsion ,General Chemistry ,Liquid waste - Published
- 2015
19. IN VITRO STUDY OF ANTAGONISTIC POTENTIAL OF SOME FUNGI AND BACTERIA AGAINST FUSARIUM MONILIFORME VAR. SUBGLUTINANS, CAUSAL ORGANISM OF MANGO PANICLE MALFORMATION
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M.A. Hasan, S.K. Ghosh, and S.K. Ray
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Horticulture ,biology ,Fusarium moniliforme var. subglutinans ,Botany ,In vitro study ,Causal organism ,biology.organism_classification ,Bacteria ,Panicle - Published
- 2014
20. INTEGRATED NUTRIENT MANAGEMENT IN IMPROVING FRUIT QUALITY OF MANGO 'HIMSAGAR'
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K. Bhattacharaya, S K Mandal, P. Dutta, Srivardhini K. Jha, S.K. Ray, M.A. Hasan, H. Banerjee, and M. Manna
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Soil bacteria ,Farmyard manure ,Nutrient management ,media_common.quotation_subject ,Horticulture ,Biology ,engineering.material ,Ascorbic acid ,Soluble solids ,engineering ,Crop quality ,Quality (business) ,Vermicompost ,media_common - Published
- 2013
21. PACLOBUTRAZOL IN MITIGATING IRREGULAR BEARING IN MANGO
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B. Singh, K.K. Mandal, S.K. Ray, S. Jha, S. Mandal, M.A. Hasan, B. C. Das, M. Manna, and S. Kole
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chemistry.chemical_compound ,Horticulture ,Bearing (mechanical) ,chemistry ,law ,Botany ,Biology ,law.invention ,Paclobutrazol - Published
- 2013
22. NUTRIENTS UPTAKE BY FRUIT AS INFLUENCED BY ORGANIC AND INORGANIC FERTILIZERS
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Srivardhini K. Jha, S. Mandal, M.A. Hasan, S.K. Ray, K. Bhattacharya, H. Banerjee, P. Dutta, and M. Manna
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Nutrient ,Agronomy ,Chemistry ,Horticulture - Published
- 2013
23. FOLIAR NUTRIENT CONTENT IN MANGO AS INFLUENCED BY ORGANIC AND INORGANIC NUTRIENTS AND THEIR CORRELATIVE RELATIONSHIP WITH YIELD AND QUALITY
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M.A. Hasan, Srivardhini K. Jha, S. Mandal, M. Manna, S.K. Ray, P. Dutta, K. Bhattacharya, and H. Banerjee
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Horticulture ,Nutrient ,Agronomy ,media_common.quotation_subject ,Yield (wine) ,Environmental science ,Quality (business) ,media_common ,Nutrient content - Published
- 2013
24. SCREENING OF BANANA CULTIVARS TO BIOTIC STRESSES
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M.A. Hasan, S. Jha, B. Ghosh, K.K. Mandal, and R.R. Choudhury
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Horticulture ,Agronomy ,Cultivar ,Biology - Published
- 2013
25. EVALUATION OF SEEDED BANANA TYPES FOR BIOTIC STRESS
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Kapil Mandal, M. Manna, R. Ray Chowdhury, S. Jha, Dipaloke Majumder, and M.A. Hasan
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Horticulture ,Black sigatoka ,Plant virus ,Botany ,Biotic stress ,Plant disease resistance ,Biology - Published
- 2011
26. STUDIES ON GENETIC DIVERGENCE OF SEEDED BANANA IN WEST BENGAL
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R. Ray Chowdhury, M. Manna, Kapil Mandal, Dipaloke Majumder, S. Jha, and M.A. Hasan
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Multivariate statistics ,Bract ,Horticulture ,Similarity (network science) ,Pollen ,Single-linkage clustering ,Dendrogram ,medicine ,medicine.disease_cause ,Parthenocarpy ,Explained variation ,Mathematics - Abstract
Thirty four (34) types of seeded banana with B genome collected from different agroclimatic zones of West Bengal, India were characterised at the Horticultural Research Station, West Bengal, India during the years 2005-2008 by using 123 plant morphological characters with different multivariate techniques. The proximity matrix, both by squared Euclidean and cophenetic correlation, between types indicated high closeness/similarity among ‘Attiakala’, ‘Bichkela-1’, ‘Bichkela-2’ and ‘Hill Banana’. The highest proximity value of 20.62 with ‘Kalyani Local-3’ showed maximum dissimilarity with ‘Maricha’ and ‘Jhama Diara’. A dendrogram using the single linkage clustering technique on squared Euclidean distance matrix and cophenetic correlation matrix showed 13 and 14 clusters, respectively. PCA was used by considering 13 factors on the basis of variance explained, i.e. more than 3%, and total explained variation was confined to 68%. Thus considering the dominant characters with positive loading under Factor 1, such as bract scar on rachis, pollen sacs colour, leaf corrugation, general fruit shape, fruits and compound tepal, the positively loaded types were ‘Bichkela-1’, ‘Hill Banana’, ‘Attiakala’, ‘Bichkela-2’, ‘Kalyani Local-1’ and ‘Jhama Daira’. Factor 1 thus explained 14.21% of the total variance. In general, PCA results agreed with the results obtained by cluster analysis. PCA gave a better picture of the relationship between seeded banana types than cluster analysis and was useful in confirming group(s) obtained through cluster analysis. MDS group plots also clearly indicated the clustering of homogeneous types. Among the 34 seeded banana types, 32 were assessed as parthenocarpic. The two non-parthenocarpic types identified were ‘Baruipur’ and ‘Seed Banana-15’.
- Published
- 2011
27. Concurrent Error Detection in Finite-Field Arithmetic Operations Using Pipelined and Systolic Architectures
- Author
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M.A. Hasan and Siavash Bayat-Sarmadi
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Polynomial ,Systolic array ,Parallel computing ,Operand ,Theoretical Computer Science ,Normal basis ,Polynomial basis ,Computational Theory and Mathematics ,Hardware and Architecture ,Multiplier (economics) ,Finite field arithmetic ,Arithmetic ,Error detection and correction ,Software ,Mathematics - Abstract
In this work, we consider detection of errors in polynomial, dual, and normal bases arithmetic operations. Error detection is performed by recomputing with the shifted operand method, while the operation unit is in use. This scheme is efficient for pipelined architectures, particularly systolic arrays. Additionally, one semisystolic multiplier for each of the polynomial, dual, type I, and type II optimal normal bases is presented. The results show that for having better or similar space and time overheads compared to a number of related previous work, the multipliers have generally a higher error-detection capability, e.g., the error-detection capability of the RESO-based scheme for single and multiple stuck-at faults in a polynomial basis multiplier is 100 percent. Finally, we also comment on how RESO can be used for concurrent error correction to deal with transient faults.
- Published
- 2009
28. Modeling of gadolinium recovery from nitrate medium with 8-hydroxyquinoline by emulsion liquid membrane
- Author
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R.F. Aglan, M.A. Hasan, and S.A. El-Reefy
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Environmental Engineering ,Health, Toxicology and Mutagenesis ,Gadolinium ,Inorganic chemistry ,chemistry.chemical_element ,Surface-Active Agents ,chemistry.chemical_compound ,Mass transfer ,Environmental Chemistry ,Waste Management and Disposal ,Nitrates ,Aqueous solution ,Chromatography ,Drop (liquid) ,Membranes, Artificial ,Permeation ,Oxyquinoline ,Pollution ,Toluene ,Membrane ,Models, Chemical ,chemistry ,Emulsion ,Emulsions - Abstract
The extraction equilibrium of Gd(III) from nitrate medium by 8-hydroxyquinoline (HOX) in toluene was studied. Liquid–liquid investigations were first carried out. Based on the equilibrium results, the extraction of Gd(III) from aqueous nitrate medium into an emulsion liquid membrane system (ELM) containing 8-hydroxyquinoline in toluene as extractant, HNO 3 as stripping solution, Span-80 as surfactant was studied. The stability of the prepared ELM was studied in terms of the degree of membrane breakage. The different parameters affecting the permeation of gadolinium (III) were also studied. A general permeation model for the recovery of Gd(III) by the selected membrane is presented. The internal mass transfer in the water in oil (W/O) emulsion drop, the external mass transfer around the drop, the rates of formation and decomposition of the complex at the external aqueous–organic interface were considered.
- Published
- 2009
29. Error Detection and Fault Tolerance in ECSM Using Input Randomization
- Author
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A. Dominguez-Oviedo and M.A. Hasan
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Triple modular redundancy ,Discrete logarithm ,Computer science ,Distributed computing ,Redundancy (engineering) ,Fault tolerance ,Electrical and Electronic Engineering ,Elliptic curve cryptography ,Error detection and correction ,Algorithm ,Fault detection and isolation ,Key size - Abstract
For some applications, elliptic curve cryptography (ECC) is an attractive choice because it achieves the same level of security with a much smaller key size in comparison with other schemes such as those that are based on integer factorization or discrete logarithm. For security reasons, especially to provide resistance against fault-based attacks, it is very important to verify the correctness of computations in ECC applications. In this paper, error-detecting and fault-tolerant elliptic curve cryptosystems are considered. Error detection may be a sufficient countermeasure for many security applications; however, fault-tolerant characteristic enables a system to perform its normal operation in spite of faults. For the purpose of detecting errors due to faults, a number of schemes and hardware structures are presented based on recomputation or parallel computation. It is shown that these structures can be used for detecting errors with a very high probability during the computation of the elliptic curve scalar multiplication (ECSM). Additionally, we show that using parallel computation along with either PV or recomputation, it is possible to have fault-tolerant structures for the ECSM. If certain conditions are met, these schemes are more efficient than others such as the well-known triple modular redundancy. Prototypes of the proposed structures for error detection and fault tolerance have been implemented, and experimental results have been presented.
- Published
- 2009
30. High-Performance Architecture of Elliptic Curve Scalar Multiplication
- Author
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M.A. Hasan and Bijan Ansari
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Discrete mathematics ,Scalar multiplication ,Theoretical Computer Science ,Elliptic curve point multiplication ,Elliptic curve ,Finite field ,Computational Theory and Mathematics ,Hardware and Architecture ,Hardware_ARITHMETICANDLOGICSTRUCTURES ,XOR gate ,Critical path method ,Software ,AND gate ,Word (computer architecture) ,Mathematics - Abstract
A high performance architecture of elliptic curve scalar multiplication based on the Montgomery ladder method over finite field GF(2m) is proposed. A pseudo-pipelined word serial finite field multiplier with word size w, suitable for the scalar multiplication is also developed. Implemented in hardware, this system performs a scalar multiplication in approximately 6lceilm/wrceil(m-1) clock cycles and the gate delay in the critical path is equal to TAND + lceillog2(w/k)rceilTXOR, where TAND and TXOR are delays due to two-input AND and XOR gates respectively and 1 les k Lt w is used to shorten the critical path.
- Published
- 2008
31. EstimatingIn VivoAirway Surface Liquid Concentration in Trials of Inhaled Antibiotics
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M.A. Hasan and Carlos F. Lange
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Pulmonary and Respiratory Medicine ,Drug ,Cystic Fibrosis ,Taurine ,medicine.drug_class ,media_common.quotation_subject ,Respiratory System ,Antibiotics ,Microbial Sensitivity Tests ,Burkholderia cepacia ,Models, Biological ,Cystic fibrosis ,chemistry.chemical_compound ,Minimum inhibitory concentration ,In vivo ,Administration, Inhalation ,Tobramycin ,medicine ,Humans ,Computer Simulation ,Pseudomonas Infections ,Pharmacology (medical) ,Particle Size ,media_common ,Aerosols ,Dose-Response Relationship, Drug ,Thiadiazines ,Inhalation ,business.industry ,Nebulizers and Vaporizers ,Sputum ,Burkholderia Infections ,Taurolidine ,medicine.disease ,Anti-Bacterial Agents ,chemistry ,Research Design ,Anesthesia ,Pseudomonas aeruginosa ,Respiratory Mechanics ,business ,medicine.drug - Abstract
Antibiotic drugs exhibit concentration dependence in their efficacy. Therefore, ensuring appropriate concentration of these drugs in the relevant body fluid is important for obtaining the desired therapeutic and physiological action. Until recently there had been no suitable method available to measure or estimate concentration of drugs in the human airways resulting from inhaled aerosols or to determine the amount of inhaled antibiotics required to ensure minimum inhibitory concentration of a drug in the airway surface liquid (ASL). In this paper a numerical method is used for estimating local concentration of inhaled pharmaceutical aerosols in different generations of the human tracheobronchial airways. The method utilizes a mathematical lung deposition model to estimate amounts of aerosols depositing in different lung generations, and a recent ASL model along with deposition results to assess the concentration of deposited drugs immediately following inhalation. Examples of concentration estimates for two case studies: one for the antibiotic tobramycin against Pseudomonas aeruginosa, and another for taurolidine against Burkholderia cepacia are presented. The aerosol characteristics, breathing pattern and properties of nebulized solutions were adopted from two recent clinical studies on efficacy of these drugs in cystic fibrosis (CF) patients and from other sources in the literature. While the clinically effective tobramycin showed a concentration higher than the required in vivo concentration, that for the ineffective taurolidine was found to be below the speculated required in vivo concentration. Results of this study thus show that the mathematical ASL model combined with the lung deposition model can be an effective tool for helping decide the optimum dosage of inhaled antibiotic drugs delivered during human clinical trials.
- Published
- 2007
32. On Concurrent Detection of Errors in Polynomial Basis Multiplication
- Author
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M.A. Hasan and Siavash Bayat-Sarmadi
- Subjects
Binary number ,Binary multiplier ,Polynomial basis ,Hardware and Architecture ,Overhead (computing) ,Multiplication ,Multiplier (economics) ,Hardware_ARITHMETICANDLOGICSTRUCTURES ,Electrical and Electronic Engineering ,Arithmetic ,Error detection and correction ,Algorithm ,Software ,Mathematics ,Parity bit - Abstract
The detection of errors in arithmetic operations is an important issue. This paper discusses the detection of multiple-bit errors due to faults in bit-serial and bit-parallel polynomial basis (PB) multipliers over binary extension fields. Our approach is based on multiple parity bits. Experimental results presented here show that due to an increase in the number of parity bits, the area overhead tends to increase linearly, but the probability of error detection approaches unity fairly quickly, e.g., for eight parity bits. In bit-serial implementation of a GF(2163) PB multiplier using eight parity bits, the area overhead and the probability of error detection are 10.29% and 0.996, respectively. This is achieved without any increase in the computation time of the GF(2163) PB multiplier
- Published
- 2007
33. STUDIES ON THE RESPONSE OF PRUNING FOR REJUVENATION OF OLD GUAVA ORCHARD
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K.K. Mandal, B. C. Das, M.A. Hasan, S. K. Sarkar, J. Basu, S. K. D. Ray, S. Kundu, B.C. Banik, and Srivardhini K. Jha
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Horticulture ,Biology ,Orchard ,Pruning ,Rejuvenation - Published
- 2007
34. A New Approach to Subquadratic Space Complexity Parallel Multipliers for Extended Binary Fields
- Author
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Haining Fan and M.A. Hasan
- Subjects
Discrete mathematics ,Polynomial ,Basis (linear algebra) ,Toeplitz matrix ,Theoretical Computer Science ,Polynomial basis ,Finite field ,Computational Theory and Mathematics ,Hardware and Architecture ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Multiplier (economics) ,Multiplication ,Circuit complexity ,Algorithm ,Software ,Mathematics - Abstract
Based on Toeplitz matrix-vector products and coordinate transformation techniques, we present a new scheme for subquadratic space complexity parallel multiplication in GF(2n) using the shifted polynomial basis. Both the space complexity and the asymptotic gate delay of the proposed multiplier are better than those of the best existing subquadratic space complexity parallel multipliers. For example, with n being a power of 2, the space complexity is about 8 percent better, while the asymptotic gate delay is about 33 percent better, respectively. Another advantage of the proposed matrix-vector product approach is that it can also be used to design subquadratic space complexity polynomial, dual, weakly dual, and triangular basis parallel multipliers. To the best of our knowledge, this is the first time that subquadratic space complexity parallel multipliers are proposed for dual, weakly dual, and triangular bases. A recursive design algorithm is also proposed for efficient construction of the proposed subquadratic space complexity multipliers. This design algorithm can be modified for the construction of most of the subquadratic space complexity multipliers previously reported in the literature
- Published
- 2007
35. Fault Detection Architectures for Field Multiplication Using Polynomial Bases
- Author
-
M.A. Hasan and Arash Reyhani-Masoleh
- Subjects
Polynomial ,Binary number ,Fault detection and isolation ,Theoretical Computer Science ,Computer Science::Hardware Architecture ,Polynomial basis ,Finite field ,Computational Theory and Mathematics ,Hardware and Architecture ,Logic gate ,Multiplication ,Hardware_ARITHMETICANDLOGICSTRUCTURES ,Error detection and correction ,Algorithm ,Software ,Mathematics - Abstract
In many cryptographic schemes, the most time consuming basic arithmetic operation is the finite field multiplication and its hardware implementation for bit parallel operation may require millions of logic gates. Some of these gates may become faulty in the field due to natural causes or malicious attacks, which may lead to the generation of erroneous outputs by the multiplier. In this paper, we propose new architectures to detect erroneous outputs caused by certain types of faults in bit-parallel and bit-serial polynomial basis multipliers over finite fields of characteristic two. In particular, parity prediction schemes are developed for detecting errors due to single and certain multiple stuck-at faults. Although the issue of detecting soft errors in registers is not considered, the proposed schemes have the advantage that they can be used with any irreducible binary polynomial chosen to define the finite field
- Published
- 2006
36. Evaluation of compressive mechanical properties of Al-foams using electrical conductivity
- Author
-
M.A. Hasan, A. Kim, S.H. Nahm, and S.S. Cho
- Subjects
Materials science ,Alloy ,Composite number ,Metal foam ,engineering.material ,Compressive strength ,Electrical resistivity and conductivity ,Powder metallurgy ,Ceramics and Composites ,engineering ,Relative density ,Composite material ,Elastic modulus ,Civil and Structural Engineering - Abstract
Al-foam filled composite panels have enormous potential for application as light weight structural members as well as the energy absorber during crushing. In such applications the mechanical properties of foams are of paramount importance. In this study, Al alloy foams of composition Al–3 wt. %Si–2%Cu–2%Mg were produced using powder metallurgical method. Mechanical properties of foams of different density were measured using the uni-axial compression test and the measured properties such as elastic modulus and strength were compared with those of other existing commercial foams. The newly produced (Al–3%Si–2%Cu–2%Mg alloy) foams showed the elastic modulus and strength similar or sometimes higher than those of the other existing commercial foams. The electrical conductivity of the foams was measured and was found to follow a power law relationship, with an exponent value of 1.5, with the relative density. A set of mathematical relationships were deduced between electrical conductivity and various compressive properties of Al-foams with an aim to propose a nondestructive method for obtaining the elastic modulus, the compressive strength and the densification strain of foams using electrical conductivity. The curves obtained from derived relationships fit very well with the experimental results.
- Published
- 2005
37. Low complexity word-level sequential normal basis multipliers
- Author
-
M.A. Hasan and Arash Reyhani-Masoleh
- Subjects
Discrete mathematics ,Field (mathematics) ,Type (model theory) ,Theoretical Computer Science ,Normal basis ,Finite field ,Computational Theory and Mathematics ,Dimension (vector space) ,Hardware and Architecture ,Multiplier (economics) ,Finite field arithmetic ,Hardware_ARITHMETICANDLOGICSTRUCTURES ,Arithmetic ,Software ,Word (computer architecture) ,Mathematics - Abstract
For efficient hardware implementation of finite field arithmetic units, the use of a normal basis is advantageous. In this paper, two classes of architectures for multipliers over the finite field GF(2/sup m/) are proposed. These multipliers are of sequential type, i.e., after receiving the coordinates of the two input field elements, they go through k, 1 /spl les/ k /spl les/ m, iterations (i.e., clock cycles) to finally yield all the coordinates of the product in parallel. The value of k depends on the word size w = /spl lceil/m/k/spl rceil/. For w > 1, these multipliers are highly area efficient and require fewer number of logic gates even when compared with the most area efficient multipliers available in the open literature. This makes the proposed multipliers suitable for applications where the value of m is large but space is of concern, e.g., resource constrained cryptographic systems. Additionally, if the field dimension m is composite, i.e., m = kn, then the extension of one class of the architectures yields a highly efficient multiplier over composite fields.
- Published
- 2005
38. Low complexity bit parallel architectures for polynomial basis multiplication over GF(2m)
- Author
-
M.A. Hasan and Arash Reyhani-Masoleh
- Subjects
Polynomial ,Irreducible polynomial ,All one polynomial ,Theoretical Computer Science ,Algebra ,Polynomial basis ,Finite field ,Computational Theory and Mathematics ,Primitive polynomial ,Hardware and Architecture ,Multiplier (economics) ,Finite field arithmetic ,Hardware_ARITHMETICANDLOGICSTRUCTURES ,Software ,Mathematics - Abstract
Representing the field elements with respect to the polynomial (or standard) basis, we consider bit parallel architectures for multiplication over the finite field GF(2m). In this effect, first we derive a new formulation for polynomial basis multiplication in terms of the reduction matrix Q. The main advantage of this new formulation is that it can be used with any field defining irreducible polynomial. Using this formulation, we then develop a generalized architecture for the multiplier and analyze the time and gate complexities of the proposed multiplier as a function of degree m and the reduction matrix Q. To the best of our knowledge, this is the first time that these complexities are given in terms of Q. Unlike most other articles on bit parallel finite field multipliers, here we also consider the number of signals to be routed in hardware implementation and we show that, compared to the well-known Mastrovito's multiplier, the proposed architecture has fewer routed signals. The proposed generalized architecture is further optimized for three special types of polynomials, namely, equally spaced polynomials, trinomials, and pentanomials. We have obtained explicit formulas and complexities of the multipliers for these three special irreducible polynomials. This makes it very easy for a designer to implement the proposed multipliers using hardware description languages like VHDL and Verilog with minimum knowledge of finite field arithmetic.
- Published
- 2004
39. Fast normal basis multiplication using general purpose processors
- Author
-
M.A. Hasan and Arash Reyhani-Masoleh
- Subjects
Flexibility (engineering) ,Multiplication algorithm ,business.industry ,Elliptic Curve Digital Signature Algorithm ,Cryptography ,Parallel computing ,Theoretical Computer Science ,Normal basis ,Finite field ,Software ,Computational Theory and Mathematics ,Hardware and Architecture ,Multiplication ,business ,Mathematics - Abstract
For cryptographic applications, normal bases have received considerable attention, especially for hardware implementation. We consider fast software algorithms for normal basis multiplication over the extended binary field GF(2/sup m/). We present a vector-level algorithm, which essentially eliminates the bit-wise inner products needed in the conventional approach to the normal basis multiplication. We then present another algorithm, which significantly reduces the dynamic instruction counts. Both algorithms utilize the full width of the data-path of the general purpose processor on which the software is to be executed. We also consider composite fields and present an algorithm, which can provide further speed-ups and an added flexibility toward hardware-software codesign of processors for very large finite fields.
- Published
- 2003
40. Effect of Azolla-urea Application on Yield and NPS Uptake by BRRI Dhan 29 in Boro Season
- Author
-
M. Jahiruddin ., M.A. Hasan ., Rehana Banu, and M.H. Mian .
- Subjects
chemistry.chemical_compound ,Agronomy ,biology ,Chemistry ,Yield (chemistry) ,Urea ,Azolla ,biology.organism_classification ,Agronomy and Crop Science ,BORO - Published
- 2003
41. Efficient multiplication beyond optimal normal bases
- Author
-
Arash Reyhani-Masoleh and M.A. Hasan
- Subjects
Discrete mathematics ,Multiplication algorithm ,Kochanski multiplication ,Field (mathematics) ,Matrix multiplication ,Theoretical Computer Science ,Normal basis ,Finite field ,Computational Theory and Mathematics ,Hardware and Architecture ,Binary operation ,Multiplication ,Software ,Mathematics - Abstract
In cryptographic applications, the use of normal bases to represent elements of the finite field GF(2/sup m/) is quite advantageous, especially for hardware implementation. In this article, we consider an important field operation, namely, multiplication which is used in many cryptographic functions. We present a class of algorithms for normal basis multiplication in GF(2/sup m/). Our proposed multiplication algorithm for composite finite fields requires a significantly lower number of bit level operations and, hence, can reduce the space complexity of cryptographic systems.
- Published
- 2003
42. Finite field multiplier using redundant representation
- Author
-
Huapeng Wu, I.F. Blake, M.A. Hasan, and Shuhong Gao
- Subjects
Very-large-scale integration ,Computational complexity theory ,Cyclic group ,Cyclotomic field ,Theoretical Computer Science ,Multiplier (Fourier analysis) ,Normal basis ,Finite field ,Computational Theory and Mathematics ,Hardware and Architecture ,Finite field arithmetic ,Arithmetic ,Algorithm ,Software ,Mathematics - Abstract
This article presents simple and highly regular architectures for finite field multipliers using a redundant representation. The basic idea is to embed a finite field into a cyclotomic ring which is based on the elegant multiplicative structure of a cyclic group. One important feature of our architectures is that they provide area-time trade-offs which enable us to implement the multipliers in a partial-parallel/hybrid fashion. This hybrid architecture has great significance in its VLSI implementation in very large fields. The squaring operation using the redundant representation is simply a permutation of the coordinates. It is shown that, when there is an optimal normal basis, the proposed bit-serial and hybrid multiplier architectures have very low space complexity. Constant multiplication is also considered and is shown to have an advantage in using the redundant representation.
- Published
- 2002
43. A new construction of Massey-Omura parallel multiplier over GF(2/sup m/)
- Author
-
Arash Reyhani-Masoleh and M.A. Hasan
- Subjects
Discrete mathematics ,Irreducible polynomial ,All one polynomial ,GF(2) ,Theoretical Computer Science ,Normal basis ,Finite field ,Computational Theory and Mathematics ,Hardware and Architecture ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Redundancy (engineering) ,Multiplier (economics) ,Circuit complexity ,Software ,Mathematics - Abstract
The Massey-Omura multiplier of GF(2/sup m/) uses a normal basis and its bit parallel version is usually implemented using m identical combinational logic blocks whose inputs are cyclically shifted from one another. In the past, it was shown that, for a class of finite fields defined by irreducible all-one polynomials, the parallel Massey-Omura multiplier had redundancy and a modified architecture of lower circuit complexity was proposed. In this article, it is shown that, not only does this type of multiplier contain redundancy in that special class of finite fields, but it also has redundancy in fields GF(2/sup m/) defined by any irreducible polynomial. By removing the redundancy, we propose a new architecture for the normal basis parallel multiplier, which is applicable to any arbitrary finite field and has significantly lower circuit complexity compared to the original Massey-Omura normal basis parallel multiplier. The proposed multiplier structure is also modular and, hence, suitable for VLSI realization. When applied to fields defined by the irreducible all-one polynomials, the multiplier's circuit complexity matches the best result available in the open literature.
- Published
- 2002
44. Efficient exponentiation using weakly dual basis
- Author
-
M.A. Hasan and Huapeng Wu
- Subjects
Modular exponentiation ,Exponentiation ,Cycles per instruction ,Fourth power ,Polynomial basis ,Finite field ,Hardware and Architecture ,Dual basis ,Electrical and Electronic Engineering ,Arithmetic ,Algorithm ,Software ,Shift register ,Mathematics - Abstract
A new architecture for finite field exponentiation using weakly dual bases is presented. An extended bidirectional linear feedback shift register is designed to multiply an arbitrary field element with certain essential multiplicands in weakly dual basis (WDB). Each of these multiplications is done in one single clock cycle. It is shown that a bit parallel implementation of the WDB fourth power has complexities comparable to those of polynomial basis fourth power. The proposed structure can effectively speed up the computation of exponentiation and is expected to reduce the power consumption compared to the conventional square and multiply scheme. Compared to the structure for polynomial basis exponentiation, the new structure is thus advantageous in a system where the WDB is already available.
- Published
- 2001
45. Low-power system-level design of VLSI packet switching fabrics
- Author
-
Amr G. Wassal and M.A. Hasan
- Subjects
Engineering ,Electronic system-level design and verification ,Design space exploration ,business.industry ,Constrained optimization ,Computer Graphics and Computer-Aided Design ,Power optimization ,Nonlinear programming ,Packet switching ,Asynchronous Transfer Mode ,Electronic engineering ,Electrical and Electronic Engineering ,business ,Architectural model ,Software - Abstract
System-level design of packet switching fabrics focuses on performance metrics and rarely considers the physical requirements that are usually addressed later at the circuit-level. However, low-power dissipation has become a major requirement in such fabrics dictated by the requirements of emerging applications and by the recent advances in fabrication and VLSI technologies. This paper proposes a framework for system-level design of packet switching fabrics that integrates performance specifications along with physical requirements and constraints. Moreover, realistic traffic models are used to derive the transition activity and the packet arrival and departure events needed for power estimation. Physical requirements are defined by an architectural model for power dissipation based on the stochastic traffic model, models for silicon area, chip count, and input-output pins, which provide a complete system-level specification of the fabric. Performance constraints are also derived from the stochastic traffic model. This framework formulates and solves the power optimization problem subject to those physical and performance constraints as an integer nonlinear optimization problem. The results obtained emphasize the importance of traffic-driven system-level optimization and show the efficiency of this framework as a system-level design space exploration tool.
- Published
- 2001
46. Look-up table-based large finite field multiplication in memory constrained cryptosystems
- Author
-
M.A. Hasan
- Subjects
Multiplication algorithm ,business.industry ,Parallel computing ,Operand ,Theoretical Computer Science ,Software ,Finite field ,Computational Theory and Mathematics ,Hardware and Architecture ,Lookup table ,Cryptosystem ,Table (database) ,Multiplication ,Arithmetic ,business ,Mathematics - Abstract
Many cryptographic systems use multiplication in the finite field GF(2/sup n/) for their underlying computations. In the recent past, a number of look-up table-based algorithms have been proposed for the software implementation of GF(2/sup n/) multiplication. Look-up table-based algorithms can provide speed advantages, but they either require a large memory space or do not fully utilize the resources of the processor on which the software is executed. In this work, an algorithm for GF(2/sup n/) multiplication is proposed which can alleviate this problem. In each iteration of the proposed algorithm, a group of bits of one of the input operands are examined and two look-up tables are accessed. The groupsize determines the table sizes, but does not affect the utilization of the processor resources. It can be used for both software and hardware realizations and is particularly suitable for implementations in memory constrained environment, such as smart cards and embedded cryptosystems.
- Published
- 2000
47. VLSI algorithms, architectures, and implementation of a versatile GF(2/sup m/) processor
- Author
-
A.G. Wassal and M.A. Hasan
- Subjects
Computational Theory and Mathematics ,Hardware and Architecture ,Software ,Theoretical Computer Science - Published
- 2000
48. Rational invariant subspace approximations with applications
- Author
-
M.A. Hasan, A.A. Hasan, and Mahmood R. Azimi-Sadjadi
- Subjects
Mathematical optimization ,Computational complexity theory ,Covariance matrix ,Signal Processing ,Invariant subspace ,Electrical and Electronic Engineering ,Linear subspace ,Algorithm ,Eigenvalues and eigenvectors ,Eigendecomposition of a matrix ,Subspace topology ,Signal subspace ,Mathematics - Abstract
Subspace methods such as MUSIC, minimum norm, and ESPRIT have gained considerable attention due to their superior performance in sinusoidal and direction-of-arrival (DOA) estimation, but they are also known to be of high computational cost. In this paper, new fast algorithms for approximating signal and noise subspaces and that do not require exact eigendecomposition are presented. These algorithms approximate the required subspace using rational and power-like methods applied to the direct data or the sample covariance matrix. Several ESPRIT- as well as MUSIC-type methods are developed based on these approximations. A substantial computational saving can be gained comparing with those associated with the eigendecomposition-based methods. These methods are demonstrated to have performance comparable to that of MUSIC yet will require fewer computations to obtain the signal subspace matrix.
- Published
- 2000
49. [Untitled]
- Author
-
Arbakariya B. Ariff, Mohamed Ismail Abdul Karim, M. Mel, and M.A. Hasan
- Subjects
Langmuir ,Chromatography ,biology ,Physiology ,Chemistry ,Rhizopus oligosporus ,Kinetics ,Biosorption ,Sorption ,General Medicine ,biology.organism_classification ,Applied Microbiology and Biotechnology ,Concentration ratio ,Adsorption ,Transmission electron microscopy ,Biotechnology ,Nuclear chemistry - Abstract
The kinetics and mechanism of lead biosorption by powderized Rhizopus oligosporus were studied using shake flask experiment. The optimum biomass concentration and initial solution pH for lead sorption at initial lead concentrations ranging from 50–200 mg/l was obtained at 0.5 g/l and pH5, respectively. In term of the ratio of initial lead concentration to biomass concentration ratio, the highest lead adsorption was obtained at 750 mg/g which gave the maximum lead uptake capacity of 126 mg/g. The experimental data of lead sorption by R.oligosporus fitted well to the Langmuir sorption isotherm model, indicating that the sorption was similar to that for an ion-exchange resin. This means that the sorption is a single layer metal adsorption that occurred as a molecular surface coverage. This assumption was confirmed by the examination of lead sorption using transmission electron microscope and energy dispersive X-ray analysis, which showed that during sorption most of the lead was adsorbed on the surface of cell.
- Published
- 1999
50. Closed-form expression for the average weight of signed-digit representations
- Author
-
Huapeng Wu and M.A. Hasan
- Subjects
Discrete mathematics ,Computational Theory and Mathematics ,Integer ,Hardware and Architecture ,Canonical form ,Closed-form expression ,Body weight ,Software ,Numerical digit ,Theoretical Computer Science ,Mathematics - Abstract
In radix-r number system, the minimal weight signed-digit (SD) representation has minimal number of nonzero signed-digits which belong to the set {/spl plusmn/1, /spl plusmn/2, ..., /spl plusmn/(r-1)}. In this article, we derive closed form expressions for the average number of nonzero digits in the minimal weight SD representation and for the average length of the canonical SD representation, a special case of the minimal weight SD form, of a positive integer whose radix-r form is of length n, n/spl ges/1.
- Published
- 1999
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