Review Article: Metal-Matrix Nano Composites, Processing .
IJSRD - International Journal for Scientific Research & Development Vol. 6, Issue 06, 2018 ISSN (online): 2321-0613Review Article: Metal-Matrix Nano Composites, Processing,Manufacturing & Application: An OverviewVijayalakshmi S.Lecturer (Senior Grade)Ayya Nadar Janaki Ammal Polytechnic College, Sivakasi, IndiaAbstract— This review is designed to be a comprehensivesource for metal matrix nano composite research, includingfundamental structure/property relationships, manufacturingtechniques and applications of metal-matrix nanocompositesmaterials. In addition to presenting the scientific frameworkfor the advances in metal- matrix nanocomposites research,this review focuses on the scientific principles andmechanism in relation to the methods of processing andmanufacturing with a discussion on commercial applicationsand health/safety concerns a critical issue for production andscale-up). Hence, this review offer a comprehensivediscussion on technology, modeling, characterization,processing, manufacturing, applications, and healthy/safetyconcerns for metal matrix nanocomposites.Key words: Nano Composites, Metal Matrix, Fabrication &ProcessingI. INTRODUCTIONNanocomposites are composites in which at least one of thephases shows dimensions in the nano meter range (1 nm 10–9 m) 1. Nanocomposite materials have emerged assuitable alternatives to overcome limitations of microcomposites and monolithic, while posing preparationchallenges related to the control of elemental compositionand stoichiometry in the nanocluster phase. They are reportedto be the materials of 21st century in the view of possessingdesign uniqueness and property combinations that are notfound in conventional composites.The general understanding of these properties is yetto be reached2, even though the first inference on them wasreported as early as 1992.As in the case of micro composites, nano compositematerials can be classified, according to their matrixmaterials, in three different categories Ceramic Matrix Nanocomposites (CMNC) Metal Matrix Nanocomposites (MMNC) Polymer Matrix Nanocomposites (PMNC)Metal matrix composites (MMCs) reinforced withnano-particles, also called Metal Matrix nano-Composites(MMnCs), and are being investigated worldwide in recentyears, owing to their promising properties suitable for a largenumber of functional and structural applications. The reducedsize of the reinforcement phase down to the nano-scale issuch that interaction of particles with dislocations becomes ofsignificant importance and, when added to otherstrengthening effects typically found in conventional MMCs,results in a remarkable improvement of mechanical properties[1–4]. The main issue to be faced in the production ofMMnCs is the low wettability of ceramic nano-particles withthe molten metal matrix, which do not allow the productionof MMnCs by conventional casting processes. Small powderaggregates are in fact prone to form clusters, losing theircapability to be homogeneously dispersed throughout thematrix for an optimal exploitation of the strengtheningpotential. For this reason, several alternative methods havebeen proposed in order to overcome this problem.Aluminium metal matrix nano composites (AlMMNCs) are a new generation of materials that have thepotential of satisfying the recent demands of advancedengineering applications. They are widely used in automobileindustries, aircrafts, structural applications and many otherdefense systems. Researchers have been observed that theaddition of nano sized SiCp particles with aluminium alloymatrix yields superior mechanical and physical properties andinterfacial characteristics of nano composites. The Scanningelectron micrographs of the Al-MMNCs indicate that thenano SiCp reinforcing particles are uniformly distributed inthe matrix alloy. This paper attempts to review the fabricationmethods and mechanical properties of Al alloy/SiCp basedmetal matrix nano composites ductility, low cost and widerange of applications. It has reduced the overall weight,pollution and fuel consumption in the aircraft andautomobiles. Alumina (Al2O3) [01], boron carbide (B4C)[02], carbon nano tubes (CNT) [03], graphite (Gr)[04],titanium dioxide (TiO2) [05] silicon carbide (SiCp) [06],tungsten carbide (WC) [07], silicon nitride (Si3N4) [08],aluminium nitride (AlN) [09], titanium carbide (TiC) [10] andsilica (SiO2) [11] are ceramic reinforcement that has beenstudied but silicon carbide and alumina are mostly utilizedcompared to other reinforcing particulates. Conventionalaluminium metal matrix composites reinforced with SiCp orAl2O3 have shown improved strength and specific stiffnessover the alloys but effected on ductility and fracturetoughness. The SiCp possesses important characteristics lowcost, improves wear behaviour, easy availability andpromotes an increase in the young’s modulus and tensilestrength of the composite. Fracture toughness and ductilityare important material properties for preventing failuresunder in service shock load applications. These havenecessitated the use of more reinforcing particulates cements in aluminium alloy metal matrix compositesare of micro level, however the technological processadvancement in nano sciences makes it possible to use nanosized reinforcement in metal matrix composites and these arecalled as Metal Matrix Nano Composites [12]. Aluminiummetal matrix composites reinforced with nano silicon carbideparticles are attractive materials for aerospace, automobileand electronic applications. Nano silicon carbidereinforcements can significantly improve mechanicalstrength, creep resistance at higher temperature, bettermachinability and higher fatigue life without affectingductility. This paper is aimed to review fabricationtechniques, mechanical properties of metal nano matrixcomposites.All rights reserved by www.ijsrd.com37
Review Article: Metal-Matrix Nano Composites, Processing, Manufacturing & Application: An Overview(IJSRD/Vol. 6/Issue 06/2018/010)II. MATERIALSMetal Matrix Composites are composed of a metallic matrix(aluminium, magnesium, iron, cobalt, copper) and a dispersedceramic (oxides, carbides) or metallic (lead, tungsten,molybdenum) phase.Several metallic materials have been considered asmatrix constituent for the preparation of MMnCs. Inparticular, the most interesting metals for industrialapplications are Al, Mg, Ti, Cu and their alloys. Pure andalloyed aluminum is the most investigated material with thelargest number of published research studies describing Albased composites as possible candidates for structuralapplication. Different species of nano-sized oxides asreinforcement agents. Especially, carborundum and aluminaare the most common ceramic reinforcements for MMNCs.Moreover, different allotropes of carbon (carbon black ,fullerenes and carbon nanotubes have been investigated asfillers for several research works published in literature. Themost used particles are CNTs: they confer very highmechanical properties to the metal matrix and, meanwhile,they lead to increased electrical conductivity, which makesMMnCs very attractive materials for electrical applications.Single wall carbon nanotubes (SWCNT) and multi wallcarbon nanotubes (MWCNT) are both used for MMnCsproduction. In this regard, for example copper-0.1 wt. %MWCNT composites revealed a 47% increase in hardnessand bronze-0.1 wt. % SWCNT showed a 20% improvedelectrical conductivity. Finally, intermetallic compounds(NiAl, Al3Ti) have also been successfully used asreinforcement phase in MMnCs. Al–Al3Ti 38anocompositesrevealed good mechanical behavior at hightemperature, whileTiAl–NiAl MMnCs showed low fracture toughness and highhardness (Al2O3, Y2O3), nitrides (Si3N4, AlN) , carbides(TiC,SiC), hydrates (TiH2) [47] and borides (TiB2) [28,51]have been employedA. Liquid State Fabrication of Metal Matrix CompositesIt involves incorporation of dispersed phase into a moltenmatrix metal, followed by its solidification. In order toprovide high level of mechanical properties of the composite,good interfacial bonding (wetting) between the dispersedphase and the liquid matrix should be obtained. Wettingimprovement may be achieved by coating the dispersed phaseparticles (fibers). Proper coating not only reduces interfacialenergy, but also prevents chemical interaction between thedispersed phase and the matrix. The methods of liquid statefabrication of Metal Matrix Composites: Stir casting,Infiltration like gas pressure infiltration, Squeeze castinginfiltration or Pressure die infiltration.1) Stir CastingIt is a liquid state method of composite materials fabrication,in which a dispersed phase (ceramic particles, short fibers) ismixed with a molten matrix metal by means of mechanicalstirring. Stir casting as shown in figure is the simplest and themost cost effective method of liquid state fabrication. Theliquid composite material is then cast by conventional castingmethods and may also be processed by conventional Metalforming technologies.III. FABRICATION TECHNIQUESProcessing of metal matrix nano composites (MMNC’s) areclassified into solid state and liquid state. Solid state includesDiffusion Bonding, Electroplating, Powder Metallurgy,Spray Deposition, Immersion Plating, Chemical VapourDeposition and Physical Vapor Deposition etc. Liquid stateprocessing includes Stir Casting, Squeeze Casting, MeltInfiltration, Compo Casting and Melt oxidation processingetc. Researchers have reported the manufacturing processessuch as Powder Metallurgy, Stir Casting, High Energy BallMilling, Squeeze Casting, Mechanical Alloying, Pressureless/Infiltration; Spark Plasma Sintering, UltrasonicCavitation’s based solidification, Vortex Process, Sol-Gelsynthesis and Laser Deposition for Al based nano composites.There are few limitations of traditional consolidationprocesses which unable to retain the nano scale grain sizeowing to the excessive grain growth during processing. Thenano particles have the tendency of agglomeration andclustering, due to high surface energy, electrostatic, moistureadhesiveness and attractive Vander Waal’s bonding, whichaffect its uniform distribution during processing. The variousfabrication methods and morphology of nano composites lasttwo decades is reviewed.Fig. 1: Stir Casting2) InfiltrationIt is a liquid state method of composite materials fabrication,in which a preformed dispersed phase (ceramic particles,fibers, woven) is soaked in a molten matrix metal, which fillsthe space between the dispersed phase inclusions.The motive force of an infiltration process may beeither capillary force of the dispersed phase (spontaneousinfiltration) or an external pressure (gaseous, mechanical,electromagnetic, centrifugal or ultrasonic) applied to theliquid matrix phase (forced infiltration).a)Gas Pressure InfiltrationIt is a forced infiltration method of liquid phase fabrication ofMetal Matrix Composites, using a pressurized gas forapplying pressure on the molten metal and forcing it topenetrate into a preformed dispersed phase as shown infigure. Gas Pressure Infiltration method is used formanufacturing large composite parts.All rights reserved by www.ijsrd.com38
Review Article: Metal-Matrix Nano Composites, Processing, Manufacturing & Application: An Overview(IJSRD/Vol. 6/Issue 06/2018/010)Fig. 2: Gas Pressurization InfiltrationSqueeze Casting Infiltration or pressure dieinfiltration – It is a forced infiltration method of liquid phasefabrication of Metal Matrix Composites, using a movablemold part (ram) for applying pressure on the molten metaland forcing it to penetrate into a preformed dispersed phase,placed into the lower fixed mold part as shown in figure. Themethod is used for manufacturing simple small parts likeautomotive engine pistons.2) Powder Processing or Powder MetallurgyThese methods in conjunction with deformation processingare used to fabricate particulate or short fiber reinforcedcomposites as shown in figure. This typically involves coldpressing and sintering, or hot pressing to fabricate primarilyparticle- or whisker-reinforced MMCs. The matrix and thereinforcement powders are blended to produce ahomogeneous distribution.The blending stage is followed by cold pressing toproduce what is called a green body, which is about 80%dense and can be easily handled. The cold pressed green bodyis canned in a sealed container and degassed to remove anyabsorbed moisture from the particle surfaces. The material ishot pressed, uniaxial or ISO-statically, to produce a fullydense composite and extruded.Fig. 5: Powder MetallurgyFig. 3: Squeeze Infiltration CastingB. Solid State Fabrication of Metal Matrix CompositesSolid state fabrication of metal matrix composites is theprocess , in which MMC are formed as a result of bonding ofmatrix metal and dispersed phase due to mutual diffusionoccurring between them in solid state at elevated temperatureand under pressure.1) Diffusion BondingIt is a common solid-state processing technique for joiningsimilar or dissimilar metals. Inter diffusion of atoms betweenclean metallic surfaces, in contact at an elevated temperature,leads to bonding. It is also used for fabrication of MMC asshown in figure the principal advantages of this technique arethe ability to process a wide variety of metal matrices andcontrol of fiber orientation and volume fraction.C. Deposition TechniquesThese processes for metal-matrix composite fabricationinvolve coating individual fibers in a tow with the matrixmaterial needed to form the composite followed by diffusionbonding to form a consolidated composite plate or structuralshape. The main disadvantage of using deposition techniquesis that they are time consuming. Several depositiontechniques are available: immersion plating, electroplating,and spray deposition, chemical vapor deposition (CVD), andphysical vapor deposition (PVD), spray forming.1) Spray FormingOne particular example of this, a co-spray process, uses aspray gun to atomize a molten aluminum alloy matrix, intowhich heated silicon carbide particles are injected as shownin figure.Fig. 5: Spray FormingFig. 4: Diffusion BondingAll rights reserved by www.ijsrd.com39
Review Article: Metal-Matrix Nano Composites, Processing, Manufacturing & Application: An Overview(IJSRD/Vol. 6/Issue 06/2018/010)Dipping or immersion plating is similar toinfiltration casting except that fiber tows are continuouslypassed through baths of molten metal, slurry, sol, ororganometallic precursors.2) ElectroplatingIt produces a coating from a 10 µm solution containing theion of the desired material in the presence of an electriccurrent. Fibres are wound on a mandrel, which serves as thecathode, and placed into the plating bath with an anode of thedesired matrix material.3) Spray DepositionThis technique typically consists of winding fibers onto a foilcoated drum and spraying molten metal onto them to form amonotype. The source of molten metal may be powder orwire feedstock which is melted in a flame, arc, or plasmatorch.4) Chemical Vapour Deposition (CVD)It is a vaporized component decomposes or reacts withanother vaporized chemical on the substrate to form a coatingon that substrate. The processing is generally carried out atelevated temperatures.D. In-situ Fabrication of Metal Matrix CompositesIn these techniques the reinforcement phase is formed in situ.The composite material is produced in one step from anappropriate starting alloy, thus avoiding the difficultiesinherent in combining the separate componentsE. Two-Phase ProcessesTwo-phase processes like osprey deposition, compo casting,etc. involve the mixing of ceramic and matrix in a region ofthe phase diagram where the matrix contains both solid andliquid phases.(thanks to their thermal properties) or as antennas (thanks totheir electrical properties and stiffness). Aerospace andautomotive industries may exploit all the above properties fordifferent kind of applications such as structural radiators,gears, aircraft fins, cylinder liners, disk brakes and , magnetic devices.Wear resistant coatings and hemical applications.2Aerospace, naval and automotiveAl/SiCstructures.Cu/AlElectronic packaging.2O3Al/AlNMicroelectronic industry.High speed machinery, tooling,Ni/TiN, Ni/ZrN,optical and magnetic storageCu/ZrNmaterials.Structural materials for highNb/Cutemperature applications.Fe/FeStructural ronic industry.2O3Microelectronics, optical devices,Au/Aglight energy conversion.Fe/MgOCatalysts, magnetic devices.Table 1: Application of Metal Nano CompositesV. CONCLUSIONIV. APPLICATIONSMetal matrix composites reinforced by nanoparticles ornanotubes are not yet being employed in relevant commercialapplications due to their very recent development. However,MMnCs show higher mechanical properties than microparticles reinforced composites, without any evidence of astrong drop in thermal and electrical conductivity.For this reason, they are considered as possiblecandidates for substituting conventional MMCs or relatedmonolithic alloys in structural and electrical RT and HTapplications. For example, CNT composites could replace,thanks to their higher strength and stiffness, carbon fiberscomposite in many applications, especially in hightemperature environments. Another good opportunity for thesubstitution of traditional MMCs with nano-sizedcounterparts is related to the loss in fracture toughness andductility occurring in micro-reinforced MMCs. Toughnesscan be substantially preserved in nano-reinforced compositesowing to the reduced particle volume fraction required toachieve strengthening.The enhanced wear resistance and the good thermalconductivity combined to the high specific strength makeMMnCs attractive materials for aircraft brakes. Moreover, thespecific strength and elastic modulus could be exploited insport industry, for instance for rackets or bicycle frames andother components. A further field of potential application isin electronic devices, for example for heat sinks and soldersIn conclusion, new technologies require materials showingnovel properties and/or improved performance compared toconventionally processed components. In this context, metalmatrix nanocomposites are suitable materials to meet theemerging demands arising from scientific and technologicadvances. Processing methods for different types of metalmatrix nanocomposites are available, but some of these posechallenges thus giving opportunities for researchers toovercome the problems being encountered with Nano sizematerials.They offer improved performance over monolithicand microcomposites counterparts and are consequentlysuitable candidates to overcome the limitations of manycurrently existing materials and devices. A number ofapplications already exist, while much potential are possiblefor these materials, which open new vistas for the future. Inview of their unique properties such as very high mechanicalproperties even at low loading of reinforcements, gas barrierand flame related properties, many potential applications andhence the market for these materials have been projected invarious sectors. Thus all the three types of nanocompositesprovide opportunities and rewards creating new world wideinterest in these new materials.All rights reserved by www.ijsrd.com40
Review Article: Metal-Matrix Nano Composites, Processing, Manufacturing & Application: An Overview(IJSRD/Vol. 6/Issue 06/2018/010)REFERENCES[1] Roy R, Roy RA, Roy DM. Alternative perspectives mposites. Materials Letters. 1986; 4(8-9):323328.[2] Schmidt D, Shah D, Giannelis EP. New advances inpolymer/layered silicate nanocomposites. CurrentOpinion in Solid State & Materials Science. 2002;6(3):205-212.[3] Gleiter H. Materials with ultrafne microstructures:retrospectives and perspectives. NanostructuredMaterials. 1992; 1(1):1-19.[4] Braun T, Schubert A, Sindelys Z. Nanoscience andnanotechnology on the balance. Scientometrics. 1997;38(2):321-325.[5] Kamigaito O. What can be improved by nanometercomposites? Journal of Japan Society of PowderMetalurgy. 1991; 38:315-321.[6] Iijima S. Helical microtubes of graphitic carbon. Nature.1991; 354(6348):56-58.[7] Biercuk MJ, Llaguno MC, Radosvljevicm, HJ. Carbonnanotube composites for thermal management. AppiedPhysics Letters. 2002; 80(15):2767-2769.[8] Ounaies Z, Park C, Wise KE, Siochi EJ, Harrison JS.Electrical properties of single wall carbon nanotubereinforced polyimide composites. Composites Scienceand Technology. 2003; 63(11):1637-1646.[9] Weisenberger MC, Grulke EA, Jacques D, Ramtell T,Andrews R. Enhanced mechanical properties ofpolyacrylonitrile: multiwall carbon nanotube compositefibers. Journal of Nanoscience and Nanotechnology.2003; 3(6):535-539.[10] Dalton AB, Coolins S, Muñoz E, Razal JM, Ebron VH,Ferraris JP et al. Super-tough carbon-nanotube fbres:these extraordinary composite fbres can be woven intoelectronic textiles. Nature. 2003; 423(6941):703-703.[11] Anuj Dixit, Kaushik Kumar. Optimization of MechanicalProperties of Silica Gel Reinforced Aluminium MMC byusing Taguchi Method. Materials Today: Proceeding2015; 2: 2359-2366.[12] S.M. Choi, H. Awaji. Nano Composite-a new materialdesign concept. Science and Technology of AdvanceMaterials 2005; 6: 2-10.[13] J. S. Sureshbabu, P. K. Nair, C. G. Kang. Fabrication andcharacterization of aluminium based nano-micro hybridmetal matrix composites. 16th International conferenceon composite materials 2007; 1-5.[14] X. P. Zhang, L. Ye, Y. W Mai, G. F. Quan, W. Wei.Investigation on diffusion bonding characteristics of SiCparticulate reinforced aluminium metal matrixcomposites (Al/SiCp-MMC). Composites Part A:applied science and manufacturing 1999; 30: 1415-1421.[15] Basak Dogru Mert. Corrosion protection of aluminum byelectrochemically synthesized composite organiccoating. Corrosion Science, G Model 2015; CS-6546: 7.[16] S. C. Tjong, Z. Y. Ma. High-temperature creep behaviourof powder metallurgy aluminium composites reinforcedwith SiC particles of various sizes. Composites Scienceand Technology 1999; 59: 1117-1125.[17] D. J. Woo, F. C. Heer, L. N. Brewer, J. P. Hooper, S.Osswald. Synthesis of nanodiamond-reinforcedaluminum metal matrix composites using cold-spraydeposition. Carbon 2015; 86: 15-25.[18] Jia-lei ZHAO, Jin-chuan JIE, Fei CHEN, Hang CHEN,Ting-ju LI, Zhi-qiang CAO. Effect of immersion Niplating on interface microstructure and mechanicalproperties of Al/Cu bimetal. Trans. Nonferrous Met. Soc.China 2014; 24: 1659-1665.[19] Jie Tang, Genlian Fan, Zhiqiang Li, Xinda Li, Run Xu,Yao Li, Di Zhang, Won-Jin Moon, Sergy DmitrievichKaloshkin. Margarita Churyukanova. Synthesis ofcarbon nanotube/aluminium composite powders bypolymer pyrolysis chemical vapour deposition. Carbon2013; 55: 202-208.[20] L. ulmer, F. Pitard, D. Poncet, O. Demolliens. Formationof Al3Ti during physical vapour deposition of titaniumon aluminium.Microelectronic Engineering 1997; 37/38:381-387.[21] Goussous, S.; Xu, W.; Xia, K. Developing aluminumnanocomposites via severe plastic deformation. J. Phys.Conf. Ser. 2010, 240, 012106.[22] Kubota, M.; Wu, X.; Xu, W.; Xia, K. Mechanicalproperties of bulk aluminium consolidated frommechanically milled particles by back pressure equalchannel angular pressing. Mat. Sci. Eng. A 2010, 527,6533–6536.[23] Tjong, S.C. Novel nanoparticle-reinforced metal matrixcomposites with enhanced mechanical properties. Adv.Eng. Mat. 2007, 9, 639–652.[24] Li, X.; Yang, Y.; Cheng, X. Ultrasonic-assistedfabrication of metal matrix composites. J. Mater. Sci.2004, 39, 3211–3212.[25] Mao, S.X.; McMinn, N.A.; Wu, N.Q. Processing andmechanical behavior of TiAl/NiAl intermetalliccomposites produced by cryogenic emchnical alloying.Mat. Sci.Eng. A 2003, A363, 275–289.[26] Mahboob, H.; Sajjadi, S.A.; Zebarjad, S.M. Syntesis ofAl-Al2O3 Nanocomposite by Mechanical Alloying andEvaluation of the Effect of Ball Milling Time on theMicrostructure and Mechanical Properties. InProceedings of International Conference on MEMS andNanotechnology (ICMN ‘08). Kuala Lumpur, Malaysia,13–15th May, 2008, pp. 240–245.[27] Gupta, M.; Lai, M.O.; Soo, C.Y. Effect of type ofprocessing on the microstructural features andmechanical properties of Al–Cu/SiC metal matrixcomposites. Mater. Sci. Eng. A 1996, 210, 114–122.[28] Lu, L.; Lai, M.O.; Su, Y.; Teo, H.L.; Feng, C.F. In situTiB2 reinforced Al alloy composites. Scripta Mater.2001, 45, 1017–1023.[29] Gupta, M.; Lai, M.O.; Boon, M.S.; Herng, N.S.Regarding the SiC particolates size associatedmicrostructural characteristics on the aging behavior ofAl-4.5 Cu metallic matrix. Mater. Res.Bull. 1998, 33,199–209.[30] Esawi, A.M.K.; Morsi, K.; Sayed, A.; Abdel Gawad, A.;Borah, P. Fabrication and properties of dispersed carbonnanotube-aluminum composites. Mater. Sci. Eng. A2009, 508, 167–173.All rights reserved by www.ijsrd.com41
Metal Matrix Nanocomposites (MMNC) Polymer Matrix Nanocomposites (PMNC) Metal matrix composites (MMCs) reinforced with nano-particles, also called Metal Matrix nano-Composites (MMnCs), and are being investigated worldwide in recent years, owing to their promising properties suitable for a large number of functional and structural applications. .
CONTENTS CONTENTS Notation and Nomenclature A Matrix A ij Matrix indexed for some purpose A i Matrix indexed for some purpose Aij Matrix indexed for some purpose An Matrix indexed for some purpose or The n.th power of a square matrix A 1 The inverse matrix of the matrix A A The pseudo inverse matrix of the matrix A (see Sec. 3.6) A1 2 The square root of a matrix (if unique), not elementwise
A Matrix A ij Matrix indexed for some purpose A i Matrix indexed for some purpose Aij Matrix indexed for some purpose An Matrix indexed for some purpose or The n.th power of a square matrix A 1 The inverse matrix of the matrix A A The pseudo inverse matrix of the matrix A (see Sec. 3.6) A1/2 The square root of a matrix (if unique), not .
CONTENTS CONTENTS Notation and Nomenclature A Matrix Aij Matrix indexed for some purpose Ai Matrix indexed for some purpose Aij Matrix indexed for some purpose An Matrix indexed for some purpose or The n.th power of a square matrix A 1 The inverse matrix of the matrix A A The pseudo inverse matrix of the matrix A (see Sec. 3.6) A1/2 The square root of a matrix (if unique), not elementwise
CONTENTS CONTENTS Notation and Nomenclature A Matrix A ij Matrix indexed for some purpose A i Matrix indexed for some purpose Aij Matrix indexed for some purpose An Matrix indexed for some purpose or The n.th power of a square matrix A 1 The inverse matrix of the matrix A A The pseudo inverse matrix of the matrix A (see Sec. 3.6) A1 2 The sq
Amendments to the Louisiana Constitution of 1974 Article I Article II Article III Article IV Article V Article VI Article VII Article VIII Article IX Article X Article XI Article XII Article XIII Article XIV Article I: Declaration of Rights Election Ballot # Author Bill/Act # Amendment Sec. Votes for % For Votes Against %
2 Connect iPod nano to a USB 3.0 port or high-power USB 2.0 port on your Mac or PC, using the cable that came with iPod nano. 3 Follow the onscreen instructions in iTunes to register iPod nano and sync iPod nano with songs from your iTunes library. If you need help using the iPod nano Setup Assistant, see Setting up iTunes syncing on page 15.
Pool Pilot Digital Nano/Nano Digital Nano Models: 75040, 75040-xx, 75041 and 75041-xx Manifolds: 75082 or 94105 Cell: RC35/22 Digital Nano Models: 75042, 75042-xx, 75043 and 75043-xx Manifold: 94106 Cells: RC35/22 or RC28 Owner's Manual Installation / Operation This manual covers the installation
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