78/3 EISCAT TECHNICAL NOTES. THE EFFECT OF ICE ON AN ANTENNA REFLECTOR T. Hagfors. KIRUNA Sweden

Size: px
Start display at page:

Download "78/3 EISCAT TECHNICAL NOTES. THE EFFECT OF ICE ON AN ANTENNA REFLECTOR T. Hagfors. KIRUNA Sweden"

Transcription

1 78/3 EISCAT TECHNICAL NOTES THE EFFECT OF ICE ON AN ANTENNA REFLECTOR T. Hagfors KIRUNA Sweden

2 EISCAT TECHNICAL Na TE No. 3 JANUARY 1978 THE EFFECT OF ICE ON AN ANTENNA REFLECTOR T. Hagfors Summary. The effect of an ice 1a ye r on an antenna reflector i s assess e e by eva1uating the phase deviation caused by an ice layer on a plane metallic ref1ector. The ice is treated as a slightly 1assy dielectric. The calculations are made for an arbitrary angle of incidence and for both principal linear polarization The resu1ts are presented in terms of curves of the phase of the ref1ections coefficients versus angle of incidence.

3 EISCAT SCIENTIFIC ASSOCIATION S KIRUNA 1. SWEDEN re U " HOPrl E_ l l"' «I fllvl IJ7llooI(lEOFVSIl.S THE EFFECT OF l CE ON AN ANTENNA REFLECTOR The presenee of ice on a n antenna reflector modifies the antenna performance in two ways. The phase of the reflected wa ve wi l l deviate f rom the desired value ; the reflection coefficient will be l e s s tha n unity if t here is appreci able loss i n the l a y e r. The f ormer of t hese t wo effects is generally the more important since ice can often be t reated as a sliqhtly lossy d i e l e c t r i c. To assess the effect we c omp ute the p lane metallic reflector covered by dielectric as s hown in Figure l. reflection coefficient of a a un iform a l o s s y z 8 8 n =1 'r (ice) 1 n(l-i Kl 2 refleclor y Figure l. Lossy dielectr ic s lab on gerfect metallic reflector. Above t he die l e c tri c s l ab we represent the field by F =A Ox - i k O(y -si n8- z c os8 ) - iko(ysine+zc os 8 ) e +Be (1 ) where A, B s ign i fy the e lectr ic f ield norma l t o o lane of i ncidence the mag netic field in t he x- d i r e c t i o n when the nolarization is o rthogonal.

4 2 - EISCAT SCIENTIFIC ASSOCIATION $ 98'01 KIRUNA t. SWEDEN TlLEI'HOI'rIE_I\f740 H l U OlonSK s In the dielectric medium we have - i k O n (1- i K) ry - s i.ne v- ec o s a " ) = C -e - i k On (l-ik) (ys1n8 l +zcos8 ') + O e (2 ) I n med i um 2 the f ields must vanish. Whe n A,B, C a nd D s ignify incidenc e a nd when A, B,C p lane of i nc i de nc e. E fie lds, E i s norm a l to the plane of and D s iqni fy B f iel ds, E l i e s i n the Since the y-variat ion must be ident ical in media (O) and ( l) we must have : n (1- ik)sin8 ' = sin S (3) We now p ut : 8 ' = Lo (4 ) and obtain : (sin 81Cosha. + i c o s 81S i nh a ) n (1-i K) = sins (5 ) Th e two conditions must be satisfied : A. (6 ) whic h means t hat (7 ) "The argument et i s propor t ional to K to firs t o rder.

5 3 - EISCAT SCIENTIFIC ASSOCIATION S KIRUNA 1, SWEDEN n UI'ttONl_' m'*3 n u x 17M QfOfYSII S ( 8) To fir s t order i n K we recqver Snell 's law : s ine l sine l = n (9 ) I n the followi ng boundary value cansideration t he y -variation can be ignored : Case l E 1 plane of incidence -, E = E x e x ( 10) In medium l we must have Ex = O at z = O. This g ive s : c = - D. (ll) At z = d continuity of Ex gives : i k On (l- i K) zc o s S ' C (e -ik On ( 1- i K) zcos8' ) - e = ikozcos8 A'e when z = d. ( 12) Co n t i nu i t y of H gives : ikon( 1-iK)zCOS 8 ' C(n(1-iK)COSe') (e -ik On (l-i K) z c o s S' + e ) ikozcos8 = cosa (Ae - i kozc os 8 Be ) whe n z = d. ( 13) S imp l ified system of equations for B and A: -l A' "O + B ' " O - CI " l - l "l ) - l T -l A " fl O - B'" C S( " l + " l ) O ( 1 4 )

6 4 - EISCAT SCIENTIFIC ASS OCIATION S 9I1 01 KIRUNA 1. SWEDEN TlUfOttOHl_'''. nux"'" OIOf'YII( I s = cose T = n(! - ik)cos6 1 - nccos8 l _ i K ) c o s e l From which we derive : G ikodg+ - i k Od G_ e + B G+e A ikodg_ - i k Od G+ G+e + G e = = Rl (15) where ( to first o rde r in K ) G+ = In 2 -sin 2 e + cose - Case 2 E I l plane of incidence - 2 Le n I n 2 -S i n 2 e ( 16) + + B = B e x x (1 7) I n med ium l we must ha v e E y = O at z = O. This means t hat o r C = D. ab x O at O äz = z = ( 18) ( 1 9 ) The two boundary c onditions at z = dbecome : ( 20) = Or : ( 21 )

7 5 - EISCAT SCIENTIFIC ASSOCIATION S 98' 01 KIRUNA t, SWEOEN TEU I'tlO"l( 0IMI0/ HU.Xl75t QEOFYI It S where G+ and G K we have : were defined above and where to first order in - i Kn 2 ( ZCOs e :+ (22) Nume ric a l e xample : For a n ice- layer K is negligible and n 2 3.0, passibly slightly larger. Computations we r e made of Rl and R 1 I for several combinations of layer thickness and angles of i ncidence. Same r e s ul t s are s hown i n F igure 2.

8 . -i 0.10 '" "c -90 o 0.08 u O 0 06 u c -60 O." "u < O -3 O 002.c o. ' O Ang le of incidence Figure 2. Phase o f r e f l e c t i o n coefficient of R (degrees ) plotted a gainst a ngle of i ncidence (deqre e s). = '" " 90 c." " - O U c 120 "u -<, 5 O O.c o. '" I 1 8 O Angle of i nc i de nc e l Figure 28. Phase o f reflection coe f f i c i e nt of Ry agains t angl e o f incidence ( degr e e s ) (degrees) plotte d

9 o occ.-<.-< 90 o..:-l ". m = O.OB 006 O U " H " "' -o.co Angle of inc i de n c e Fig ure 2C. Phase "error" of the phase difference between R n and Rl. I plotted against angle o f i nc i de nc e (degrees). l '- - _.

10 EISCAT SCIENTIFIC ASSOCIATION S KIRUNA 1. SWEDEN TELEPHONE 0980/ TELEX 8754 GEOFYSK S

P a g e 3 6 of R e p o r t P B 4 / 0 9

P a g e 3 6 of R e p o r t P B 4 / 0 9 P a g e 3 6 of R e p o r t P B 4 / 0 9 p r o t e c t h um a n h e a l t h a n d p r o p e r t y fr om t h e d a n g e rs i n h e r e n t i n m i n i n g o p e r a t i o n s s u c h a s a q u a r r y. J

More information

P a g e 5 1 of R e p o r t P B 4 / 0 9

P a g e 5 1 of R e p o r t P B 4 / 0 9 P a g e 5 1 of R e p o r t P B 4 / 0 9 J A R T a l s o c o n c l u d e d t h a t a l t h o u g h t h e i n t e n t o f N e l s o n s r e h a b i l i t a t i o n p l a n i s t o e n h a n c e c o n n e

More information

T h e C S E T I P r o j e c t

T h e C S E T I P r o j e c t T h e P r o j e c t T H E P R O J E C T T A B L E O F C O N T E N T S A r t i c l e P a g e C o m p r e h e n s i v e A s s es s m e n t o f t h e U F O / E T I P h e n o m e n o n M a y 1 9 9 1 1 E T

More information

A L A BA M A L A W R E V IE W

A L A BA M A L A W R E V IE W A L A BA M A L A W R E V IE W Volume 52 Fall 2000 Number 1 B E F O R E D I S A B I L I T Y C I V I L R I G HT S : C I V I L W A R P E N S I O N S A N D TH E P O L I T I C S O F D I S A B I L I T Y I N

More information

Beechwood Music Department Staff

Beechwood Music Department Staff Beechwood Music Department Staff MRS SARAH KERSHAW - HEAD OF MUSIC S a ra h K e rs h a w t r a i n e d a t t h e R oy a l We ls h C o l le g e of M u s i c a n d D ra m a w h e re s h e ob t a i n e d

More information

Software Process Models there are many process model s in th e li t e ra t u re, s om e a r e prescriptions and some are descriptions you need to mode

Software Process Models there are many process model s in th e li t e ra t u re, s om e a r e prescriptions and some are descriptions you need to mode Unit 2 : Software Process O b j ec t i ve This unit introduces software systems engineering through a discussion of software processes and their principal characteristics. In order to achieve the desireable

More information

176 5 t h Fl oo r. 337 P o ly me r Ma te ri al s

176 5 t h Fl oo r. 337 P o ly me r Ma te ri al s A g la di ou s F. L. 462 E l ec tr on ic D ev el op me nt A i ng er A.W.S. 371 C. A. M. A l ex an de r 236 A d mi ni st ra ti on R. H. (M rs ) A n dr ew s P. V. 326 O p ti ca l Tr an sm is si on A p ps

More information

Executive Committee and Officers ( )

Executive Committee and Officers ( ) Gifted and Talented International V o l u m e 2 4, N u m b e r 2, D e c e m b e r, 2 0 0 9. G i f t e d a n d T a l e n t e d I n t e r n a t i o n a2 l 4 ( 2), D e c e m b e r, 2 0 0 9. 1 T h e W o r

More information

opposite hypotenuse adjacent hypotenuse opposite adjacent adjacent opposite hypotenuse hypotenuse opposite

opposite hypotenuse adjacent hypotenuse opposite adjacent adjacent opposite hypotenuse hypotenuse opposite 5 TRtGOhiOAMTRiC WNCTIONS D O E T F F R F l I F U A R N G TO N I l O R C G T N I T Triangle ABC bas a right angle (9Oo) at C and sides of length u, b, c. The trigonometric functions of angle A are defined

More information

Get Started on CreateSpace

Get Started on CreateSpace {paperback self publishing process} Account Set Up Go to CreateSpace.com Click Sign up Fill out Create Account info fields * Under What type of media you are considering punlishing?, you may want to choose

More information

Table of C on t en t s Global Campus 21 in N umbe r s R e g ional Capac it y D e v e lopme nt in E-L e ar ning Structure a n d C o m p o n en ts R ea

Table of C on t en t s Global Campus 21 in N umbe r s R e g ional Capac it y D e v e lopme nt in E-L e ar ning Structure a n d C o m p o n en ts R ea G Blended L ea r ni ng P r o g r a m R eg i o na l C a p a c i t y D ev elo p m ent i n E -L ea r ni ng H R K C r o s s o r d e r u c a t i o n a n d v e l o p m e n t C o p e r a t i o n 3 0 6 0 7 0 5

More information

Agenda Rationale for ETG S eek ing I d eas ETG fram ew ork and res u lts 2

Agenda Rationale for ETG S eek ing I d eas ETG fram ew ork and res u lts 2 Internal Innovation @ C is c o 2 0 0 6 C i s c o S y s t e m s, I n c. A l l r i g h t s r e s e r v e d. C i s c o C o n f i d e n t i a l 1 Agenda Rationale for ETG S eek ing I d eas ETG fram ew ork

More information

Module 5 : Plane Waves at Media Interface. Lecture 36 : Reflection & Refraction from Dielectric Interface (Contd.) Objectives

Module 5 : Plane Waves at Media Interface. Lecture 36 : Reflection & Refraction from Dielectric Interface (Contd.) Objectives Objectives In this course you will learn the following Reflection and Refraction with Parallel Polarization. Reflection and Refraction for Normal Incidence. Lossy Media Interface. Reflection and Refraction

More information

THIS PAGE DECLASSIFIED IAW E

THIS PAGE DECLASSIFIED IAW E THS PAGE DECLASSFED AW E0 2958 BL K THS PAGE DECLASSFED AW E0 2958 THS PAGE DECLASSFED AW E0 2958 B L K THS PAGE DECLASSFED AW E0 2958 THS PAGE DECLASSFED AW EO 2958 THS PAGE DECLASSFED AW EO 2958 THS

More information

Geometric Predicates P r og r a m s need t o t es t r ela t ive p os it ions of p oint s b a s ed on t heir coor d ina t es. S im p le exa m p les ( i

Geometric Predicates P r og r a m s need t o t es t r ela t ive p os it ions of p oint s b a s ed on t heir coor d ina t es. S im p le exa m p les ( i Automatic Generation of SS tag ed Geometric PP red icates Aleksandar Nanevski, G u y B lello c h and R o b ert H arp er PSCICO project h ttp: / / w w w. cs. cm u. ed u / ~ ps ci co Geometric Predicates

More information

I M P O R T A N T S A F E T Y I N S T R U C T I O N S W h e n u s i n g t h i s e l e c t r o n i c d e v i c e, b a s i c p r e c a u t i o n s s h o

I M P O R T A N T S A F E T Y I N S T R U C T I O N S W h e n u s i n g t h i s e l e c t r o n i c d e v i c e, b a s i c p r e c a u t i o n s s h o I M P O R T A N T S A F E T Y I N S T R U C T I O N S W h e n u s i n g t h i s e l e c t r o n i c d e v i c e, b a s i c p r e c a u t i o n s s h o u l d a l w a y s b e t a k e n, i n c l u d f o l

More information

I N A C O M P L E X W O R L D

I N A C O M P L E X W O R L D IS L A M I C E C O N O M I C S I N A C O M P L E X W O R L D E x p l o r a t i o n s i n A g-b eanste d S i m u l a t i o n S a m i A l-s u w a i l e m 1 4 2 9 H 2 0 0 8 I s l a m i c D e v e l o p m e

More information

WELCOME. O ne Vi si on Photography i s a award wi nni ng wed d i ng photographer & wed d i ng vi d eography i n S outh Wal e s

WELCOME. O ne Vi si on Photography i s a award wi nni ng wed d i ng photographer & wed d i ng vi d eography i n S outh Wal e s WELCOME HELLO O ne Vi si on Photography i s a award wi nni ng wed d i ng photographer & wed d i ng vi d eography i n S outh Wal e s I t і ѕ а k nоwn f ac t thаt i mages speak а thousand word аnd thаt і

More information

M M 3. F orc e th e insid e netw ork or p rivate netw ork traffic th rough th e G RE tunnel using i p r ou t e c ommand, fol l ow ed b y th e internal

M M 3. F orc e th e insid e netw ork or p rivate netw ork traffic th rough th e G RE tunnel using i p r ou t e c ommand, fol l ow ed b y th e internal C i s c o P r o f i l e C o n t a c t s & F e e d b a c k H e l p C isc o S M B S up p ort A ssistant Pass Routing Information over IPsec VPN Tunnel between two ASA/PIX H ome > W ork W ith M y S ec urity

More information

APPLICATION INSTRUCTIONS FOR THE

APPLICATION INSTRUCTIONS FOR THE APPLICATION INSTRUCTIONS FOR THE SNA KES A ND LA DDERS A DD O N FO R USE WITH THE C HESS, C HEC KERS & BO RDERS LARG E AND NUMBER SET Pro duc t Numb e r: 17-2W-062 SIZE: a ppro xima te ly 19 fe e t e a

More information

Trade Patterns, Production networks, and Trade and employment in the Asia-US region

Trade Patterns, Production networks, and Trade and employment in the Asia-US region Trade Patterns, Production networks, and Trade and employment in the Asia-U region atoshi Inomata Institute of Developing Economies ETRO Development of cross-national production linkages, 1985-2005 1985

More information

o Alphabet Recitation

o Alphabet Recitation Letter-Sound Inventory (Record Sheet #1) 5-11 o Alphabet Recitation o Alphabet Recitation a b c d e f 9 h a b c d e f 9 h j k m n 0 p q k m n 0 p q r s t u v w x y z r s t u v w x y z 0 Upper Case Letter

More information

Second Order Linear Equations

Second Order Linear Equations Second Order Linear Equations Linear Equations The most general linear ordinary differential equation of order two has the form, a t y t b t y t c t y t f t. 1 We call this a linear equation because the

More information

7.1 THE MAGNETIC FIELD

7.1 THE MAGNETIC FIELD 7.1 THE MAGNETIC IEL - If one pours small iron particles around a permanent magnet ( natural lodestones or manmade ) an ordered pattern appears(fig.1). One can easily discern the presence of lines that

More information

APPLICATION INSTRUCTIONS FOR THE

APPLICATION INSTRUCTIONS FOR THE APPLICATION INSTRUCTIONS FOR THE USA MA P WITH C A PITA LS & O C EA NS (LA RG E) Pro duc t Numbe r: 16-13W-051 SIZE: a ppro xima te ly 24 fe e t wide x 15 fe e t lo ng (ma inla nd o nly) ESTIMA TED PA

More information

Dangote Flour Mills Plc

Dangote Flour Mills Plc SUMMARY OF OFFER Opening Date 6 th September 27 Closing Date 27 th September 27 Shares on Offer 1.25bn Ord. Shares of 5k each Offer Price Offer Size Market Cap (Post Offer) Minimum Offer N15. per share

More information

Ranking accounting, banking and finance journals: A note

Ranking accounting, banking and finance journals: A note MPRA Munich Personal RePEc Archive Ranking accounting, banking and finance ournals: A note George Halkos and Nickolaos Tzeremes University of Thessaly, Department of Economics January 2012 Online at https://mpra.ub.uni-muenchen.de/36166/

More information

Instruction Sheet COOL SERIES DUCT COOL LISTED H NK O. PR D C FE - Re ove r fro e c sed rea. I Page 1 Rev A

Instruction Sheet COOL SERIES DUCT COOL LISTED H NK O. PR D C FE - Re ove r fro e c sed rea. I Page 1 Rev A Instruction Sheet COOL SERIES DUCT COOL C UL R US LISTED H NK O you or urc s g t e D C t oroug y e ore s g / as e OL P ea e rea g product PR D C FE RES - Re ove r fro e c sed rea t m a o se e x o duct

More information

Display the following sentence: The principal said she would never agree to an ice cream social, but then she. Students play.

Display the following sentence: The principal said she would never agree to an ice cream social, but then she. Students play. 1 Lesson Plan Angle Measures Age group: 4 t h Grade, 5 t h Grade Mathematics Syllabus Primary : 4.G.A.2, 4.G.A.4, 5.F M - G.A.2, 5.F M - G.A.4 Online resources: What ' s Yo ur Angl e? Opening Teacher present

More information

d 1941. ncies ation thereors as ence, is, which r he the of are anvass ve ining, ies a study ely, fully ta special e rested gencies that are affiliated mably, these agenand at least somewhat l, they are

More information

Math 124B January 31, 2012

Math 124B January 31, 2012 Math 124B January 31, 212 Viktor Grigoryan 7 Inhomogeneous boundary vaue probems Having studied the theory of Fourier series, with which we successfuy soved boundary vaue probems for the homogeneous heat

More information

Math Assignment. Dylan Zwick. Spring Section 2.4-1, 5, 9, 26, 30 Section 3.1-1, 16, 18, 24, 39 Section 3.2-1, 10, 16, 24, 31

Math Assignment. Dylan Zwick. Spring Section 2.4-1, 5, 9, 26, 30 Section 3.1-1, 16, 18, 24, 39 Section 3.2-1, 10, 16, 24, 31 Math 2280 - Assignment 4 Dylan Zwick Spring 2013 Section 2.4-1, 5, 9, 26, 30 Section 3.1-1, 16, 18, 24, 39 Section 3.2-1, 10, 16, 24, 31 1 -z Section 2.4 - Method Numerical Approximation: Euler s 2.4.1

More information

OH BOY! Story. N a r r a t iv e a n d o bj e c t s th ea t e r Fo r a l l a g e s, fr o m th e a ge of 9

OH BOY! Story. N a r r a t iv e a n d o bj e c t s th ea t e r Fo r a l l a g e s, fr o m th e a ge of 9 OH BOY! O h Boy!, was or igin a lly cr eat ed in F r en ch an d was a m a jor s u cc ess on t h e Fr en ch st a ge f or young au di enc es. It h a s b een s een by ap pr ox i ma t ely 175,000 sp ect at

More information

I n t e r n a t i o n a l E l e c t r o n i c J o u r n a l o f E l e m e n t a r y E.7 d u, c ai ts is ou n e, 1 V3 1o-2 l6, I n t h i s a r t

I n t e r n a t i o n a l E l e c t r o n i c J o u r n a l o f E l e m e n t a r y E.7 d u, c ai ts is ou n e, 1 V3 1o-2 l6, I n t h i s a r t I n t e r n a t i o n a l E l e c t r o n i c J o ue rlne am l e not fa r y E d u c a t i o n, 2 0 1 4, 1 37-2 ( 16 ). H o w R e a d i n g V o l u m e A f f e c t s b o t h R e a d i n g F l u e n c y

More information

, > --1. [P(" )(x) (1 x) (1 x)dx. ( )(0) t( )P, (cos 0)(sin 0/2)+1/2(cos 0/2) +1/ A TRANSPLANTATION THEOREM FOR JACOBI SERIES

, > --1. [P( )(x) (1 x) (1 x)dx. ( )(0) t( )P, (cos 0)(sin 0/2)+1/2(cos 0/2) +1/ A TRANSPLANTATION THEOREM FOR JACOBI SERIES A TRANSPLANTATION THEOREM FOR JACOBI SERIES BY 1. Introduction Let P( )(x) be the Jacobi polynomial of degree n, order (a, ), defined by - [P(" )(x) (1 x) (1 x)dx (1 x)"(1 + x)p(" )(x) (-1) a [(1 x)+"(1

More information

APPLICATION INSTRUC TIONS FOR THE

APPLICATION INSTRUC TIONS FOR THE APPLICATION INSTRUC TIONS FOR THE DAISY HOPSCOTCH Pro duc t Numb e r: 12-2W-03 SIZE: a p p ro xima te ly 14 fe e t hig h x 4 fe e t wid e. ESTIMATED PAINT TIME: a p p ro xima te ly 1 2 p e o p le fo r

More information

Use precise language and domain-specific vocabulary to inform about or explain the topic. CCSS.ELA-LITERACY.WHST D

Use precise language and domain-specific vocabulary to inform about or explain the topic. CCSS.ELA-LITERACY.WHST D Lesson seven What is a chemical reaction? Science Constructing Explanations, Engaging in Argument and Obtaining, Evaluating, and Communicating Information ENGLISH LANGUAGE ARTS Reading Informational Text,

More information

APPLICATION INSTRUC TIONS FOR THE

APPLICATION INSTRUC TIONS FOR THE APPLICATION INSTRUC TIONS FOR THE CHESS/ CHECKERBOARD Using the Che ss, Che c ke rs a nd Bo rd e rs: (SMALL), Pro duc t Numb e r: 11-1W-009 SIZE: a p p ro xima te ly 10 fe e t e a c h sid e ESTIMATED PAINT

More information

Plane Trusses Trusses

Plane Trusses Trusses TRUSSES Plane Trusses Trusses- It is a system of uniform bars or members (of various circular section, angle section, channel section etc.) joined together at their ends by riveting or welding and constructed

More information

Preview from Notesale.co.uk Page 2 of 42

Preview from Notesale.co.uk Page 2 of 42 . CONCEPTS & FORMULAS. INTRODUCTION Radian The angle subtended at centre of a circle by an arc of length equal to the radius of the circle is radian r o = o radian r r o radian = o = 6 Positive & Negative

More information

Ronda Lights. D e s k. S t a n d a r d L i g h t p lo t. S e e p ag e 3. ( s o f t w a re v ersion ) g r a n d MA 2 F a d e r W ing

Ronda Lights. D e s k. S t a n d a r d L i g h t p lo t. S e e p ag e 3. ( s o f t w a re v ersion ) g r a n d MA 2 F a d e r W ing Lights D e s k 1x 1x Ma-lighting g r a n d MA 2 Light ( s o f t w a re v ersion 3.3.4.3 ) g r a n d MA 2 F a d e r W ing D i m m e r s 112x Ma d i m MA c h a n n e l 2,3kW on H a r t i n g 16 in g r id

More information

Lesson Ten. What role does energy play in chemical reactions? Grade 8. Science. 90 minutes ENGLISH LANGUAGE ARTS

Lesson Ten. What role does energy play in chemical reactions? Grade 8. Science. 90 minutes ENGLISH LANGUAGE ARTS Lesson Ten What role does energy play in chemical reactions? Science Asking Questions, Developing Models, Investigating, Analyzing Data and Obtaining, Evaluating, and Communicating Information ENGLISH

More information

CATAVASII LA NAȘTEREA DOMNULUI DUMNEZEU ȘI MÂNTUITORULUI NOSTRU, IISUS HRISTOS. CÂNTAREA I-A. Ήχος Πα. to os se e e na aș te e e slă ă ă vi i i i i

CATAVASII LA NAȘTEREA DOMNULUI DUMNEZEU ȘI MÂNTUITORULUI NOSTRU, IISUS HRISTOS. CÂNTAREA I-A. Ήχος Πα. to os se e e na aș te e e slă ă ă vi i i i i CATAVASII LA NAȘTEREA DOMNULUI DUMNEZEU ȘI MÂNTUITORULUI NOSTRU, IISUS HRISTOS. CÂNTAREA I-A Ήχος α H ris to os s n ș t slă ă ă vi i i i i ți'l Hris to o os di in c ru u uri, în tâm pi i n ți i'l Hris

More information

ARC 202L. Not e s : I n s t r u c t o r s : D e J a r n e t t, L i n, O r t e n b e r g, P a n g, P r i t c h a r d - S c h m i t z b e r g e r

ARC 202L. Not e s : I n s t r u c t o r s : D e J a r n e t t, L i n, O r t e n b e r g, P a n g, P r i t c h a r d - S c h m i t z b e r g e r ARC 202L C A L I F O R N I A S T A T E P O L Y T E C H N I C U N I V E R S I T Y D E P A R T M E N T O F A R C H I T E C T U R E A R C 2 0 2 L - A R C H I T E C T U R A L S T U D I O W I N T E R Q U A

More information

c. What is the average rate of change of f on the interval [, ]? Answer: d. What is a local minimum value of f? Answer: 5 e. On what interval(s) is f

c. What is the average rate of change of f on the interval [, ]? Answer: d. What is a local minimum value of f? Answer: 5 e. On what interval(s) is f Essential Skills Chapter f ( x + h) f ( x ). Simplifying the difference quotient Section. h f ( x + h) f ( x ) Example: For f ( x) = 4x 4 x, find and simplify completely. h Answer: 4 8x 4 h. Finding the

More information

Use precise language and domain-specific vocabulary to inform about or explain the topic. CCSS.ELA-LITERACY.WHST D

Use precise language and domain-specific vocabulary to inform about or explain the topic. CCSS.ELA-LITERACY.WHST D Lesson eight What are characteristics of chemical reactions? Science Constructing Explanations, Engaging in Argument and Obtaining, Evaluating, and Communicating Information ENGLISH LANGUAGE ARTS Reading

More information

Waves on 2 and 3 dimensional domains

Waves on 2 and 3 dimensional domains Chapter 14 Waves on 2 and 3 dimensional domains We now turn to the studying the initial boundary value problem for the wave equation in two and three dimensions. In this chapter we focus on the situation

More information

INTERIM MANAGEMENT REPORT FIRST HALF OF 2018

INTERIM MANAGEMENT REPORT FIRST HALF OF 2018 INTERIM MANAGEMENT REPORT FIRST HALF OF 2018 F r e e t r a n s l a t ion f r o m t h e o r ig ina l in S p a n is h. I n t h e e v e n t o f d i s c r e p a n c y, t h e Sp a n i s h - la n g u a g e v

More information

EQUIVALENT DIELECTRIC CONSTANT OF PERIODIC MEDIA

EQUIVALENT DIELECTRIC CONSTANT OF PERIODIC MEDIA EECS 730, Winter 2009 c K. Sarabandi EQUIVALENT DIELECTRIC CONSTANT OF PERIODIC MEDIA The behavior of electromagnetic waves can be controlled by controlling the constitutive parameters of the medium in

More information

Math 205b Homework 2 Solutions

Math 205b Homework 2 Solutions Math 5b Homework Solutions January 5, 5 Problem (R-S, II.) () For the R case, we just expand the right hand side and use the symmetry of the inner product: ( x y x y ) = = ((x, x) (y, y) (x, y) (y, x)

More information

EE/Ge 157 a SYSTEMS FOR REMOTE SENSING FROM SPACE Week 3: Visible and Near IR 3-1

EE/Ge 157 a SYSTEMS FOR REMOTE SENSING FROM SPACE Week 3: Visible and Near IR 3-1 EE/Ge 157 a SYSTEMS FOR REMOTE SENSING FROM SPACE Week 3: Visible and Near IR 3-1 TOPICS TO BE COVERED Source Characteristics Interaction Mechanisms Reflection, Scattering, Diffraction Gratings, Vibrational

More information

Nonhomogeneous Linear Differential Equations with Constant Coefficients - (3.4) Method of Undetermined Coefficients

Nonhomogeneous Linear Differential Equations with Constant Coefficients - (3.4) Method of Undetermined Coefficients Nonhomogeneous Linear Differential Equations with Constant Coefficients - (3.4) Method of Undetermined Coefficients Consider an nth-order nonhomogeneous linear differential equation with constant coefficients:

More information

Ne a l Mc lo ug hlin - Alb e rta Ag / Fo r Mike Fla nnig a n Uo fa Elle n Ma c d o na ld Uo fa

Ne a l Mc lo ug hlin - Alb e rta Ag / Fo r Mike Fla nnig a n Uo fa Elle n Ma c d o na ld Uo fa Using histo ric la nd sc a p e ve g e ta tio n struc ture fo r e c o lo g ic a l re sto ra tio n: e ffe c ts o n b urn p ro b a b ility in the Bo b Cre e k Wild la nd, Alb e rta, Ca na d a. Chris Sto c

More information

H STO RY OF TH E SA NT

H STO RY OF TH E SA NT O RY OF E N G L R R VER ritten for the entennial of th e Foundin g of t lair oun t y on ay 8 82 Y EEL N E JEN K RP O N! R ENJ F ] jun E 3 1 92! Ph in t ed b y h e t l a i r R ep u b l i c a n O 4 1922

More information

Electromagnetic Waves

Electromagnetic Waves Physics 8 Electromagnetic Waves Overview. The most remarkable conclusion of Maxwell s work on electromagnetism in the 860 s was that waves could exist in the fields themselves, traveling with the speed

More information

Waves. Daniel S. Weile. ELEG 648 Waves. Department of Electrical and Computer Engineering University of Delaware. Plane Waves Reflection of Waves

Waves. Daniel S. Weile. ELEG 648 Waves. Department of Electrical and Computer Engineering University of Delaware. Plane Waves Reflection of Waves Waves Daniel S. Weile Department of Electrical and Computer Engineering University of Delaware ELEG 648 Waves Outline Outline Introduction Let s start by introducing simple solutions to Maxwell s equations

More information

Results as of 30 September 2018

Results as of 30 September 2018 rt Results as of 30 September 2018 F r e e t r a n s l a t ion f r o m t h e o r ig ina l in S p a n is h. I n t h e e v e n t o f d i s c r e p a n c y, t h e Sp a n i s h - la n g u a g e v e r s ion

More information

Chapter 9 WAVES IN COLD MAGNETIZED PLASMA. 9.1 Introduction. 9.2 The Wave Equation

Chapter 9 WAVES IN COLD MAGNETIZED PLASMA. 9.1 Introduction. 9.2 The Wave Equation Chapter 9 WAVES IN COLD MAGNETIZED PLASMA 9.1 Introduction For this treatment, we will regard the plasma as a cold magnetofluid with an associated dielectric constant. We then derive a wave equation using

More information

MA 201: Partial Differential Equations Lecture - 11

MA 201: Partial Differential Equations Lecture - 11 MA 201: Partia Differentia Equations Lecture - 11 Heat Equation Heat conduction in a thin rod The IBVP under consideration consists of: The governing equation: u t = αu xx, (1) where α is the therma diffusivity.

More information

MMBD4448W6 SURFACE MOUNT FAST SWITCHING DIODE ARRAY FEATURES. MAXIMUM RATINGS ( TA = 25 O C unless otherwise specified ) THERMAL CHARACTERISTICS

MMBD4448W6 SURFACE MOUNT FAST SWITCHING DIODE ARRAY FEATURES. MAXIMUM RATINGS ( TA = 25 O C unless otherwise specified ) THERMAL CHARACTERISTICS SURFAE MOUNT FAST SWITHING DIODE ARRAY FEATURES Fast Switching Speed UltraSmall Surface Mount Package For General Purpose Switching Applications High onductance 0.119(3.00) 0.110(2.80) 0.075(1.90) BS 0.009(0.22)

More information

Physics 3323, Fall 2014 Problem Set 13 due Friday, Dec 5, 2014

Physics 3323, Fall 2014 Problem Set 13 due Friday, Dec 5, 2014 Physics 333, Fall 014 Problem Set 13 due Friday, Dec 5, 014 Reading: Finish Griffiths Ch. 9, and 10..1, 10.3, and 11.1.1-1. Reflecting on polarizations Griffiths 9.15 (3rd ed.: 9.14). In writing (9.76)

More information

GROWMARK, INC 2200 Sout h Ave nue, P. O. Box 587, Counc i l Bl uf f s, I A

GROWMARK, INC 2200 Sout h Ave nue, P. O. Box 587, Counc i l Bl uf f s, I A UPDATED: J ul y 8, 2004 MATERIAL SAFETY DATA SHEET GROWMARK, INC 2200 Sout h Ave nue, P. O. Box 587, Counc i l Bl uf f s, I A 51502-0587 EMERGENCY TELEPHONE NUMBER: 1- ( 800) - 798-6457 ******************

More information

Welcome to the Public Meeting Red Bluff Road from Kirby Boulevard to State Highway 146 Harris County, Texas CSJ No.: December 15, 2016

Welcome to the Public Meeting Red Bluff Road from Kirby Boulevard to State Highway 146 Harris County, Texas CSJ No.: December 15, 2016 Welcome to the Public Meeting Red Bluff Road from Kirby Boulevard to State Highway 146 Harris County, Texas CSJ No.: 0912-72-340 December 15, 2016 No formal presentation will be made. Seabrook Intermediate

More information

M Line Card Redundancy with Y-Cab l es Seamless Line Card Failover Solu t ion f or Line Card H ardw or Sof t w are Failu res are Leverages hardware Y-

M Line Card Redundancy with Y-Cab l es Seamless Line Card Failover Solu t ion f or Line Card H ardw or Sof t w are Failu res are Leverages hardware Y- Line Card Redundancy with Y-Cab l es Technical Overview 1 M Line Card Redundancy with Y-Cab l es Seamless Line Card Failover Solu t ion f or Line Card H ardw or Sof t w are Failu res are Leverages hardware

More information

shhgs@wgqqh.com chinapub 2002 7 Bruc Eckl 1000 7 Bruc Eckl 1000 Th gnsis of th computr rvolution was in a machin. Th gnsis of our programming languags thus tnds to look lik that Bruc machin. 10 7 www.wgqqh.com/shhgs/tij.html

More information

Strauss PDEs 2e: Section Exercise 4 Page 1 of 6

Strauss PDEs 2e: Section Exercise 4 Page 1 of 6 Strauss PDEs 2e: Section 5.3 - Exercise 4 Page of 6 Exercise 4 Consider the problem u t = ku xx for < x < l, with the boundary conditions u(, t) = U, u x (l, t) =, and the initial condition u(x, ) =, where

More information

Nanomaterials for Photovoltaics (v11) 11 Quantum Confinement. Quantum confinement (I) The bandgap of a nanoparticle increases as the size decreases.

Nanomaterials for Photovoltaics (v11) 11 Quantum Confinement. Quantum confinement (I) The bandgap of a nanoparticle increases as the size decreases. Quantum Confinement Quantum confinement (I) The bandgap of a nanoparticle increases as the size decreases. A rule of thumb is E g d where d is the particle size. Quantum confinement (II) The Schrodinger

More information

Separation of Variables and a Spherical Shell with Surface Charge

Separation of Variables and a Spherical Shell with Surface Charge Separation of Variabes and a Spherica She with Surface Charge In cass we worked out the eectrostatic potentia due to a spherica she of radius R with a surface charge density σθ = σ cos θ. This cacuation

More information

I. Zusammenfassung.

I. Zusammenfassung. 1038 TABELLE VI. Die mittlere Kurve. Phase v I1 P hase I v I1 Phase 1 v Ph ase I v I Phase I v m - lood 10.31 - + + 140d 6.66 1 + 230 6. 51 20d 1 13~ 19 1 60d I ll ~ 10 m - 90 10.77 _. 10 13. 32 + 70 10.

More information

ELEMENTARY LINEAR ALGEBRA

ELEMENTARY LINEAR ALGEBRA ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Second Online Version, December 1998 Comments to the author at krm@maths.uq.edu.au Contents 1 LINEAR EQUATIONS

More information

Fr anchi s ee appl i cat i on for m

Fr anchi s ee appl i cat i on for m Other Fr anchi s ee appl i cat i on for m Kindly fill in all the applicable information in the spaces provided and submit to us before the stipulated deadline. The information you provide will be held

More information

Soil moisture impact on refectance of bare soils in the optical domain [ µm]

Soil moisture impact on refectance of bare soils in the optical domain [ µm] 2 nd Workshop on Remote Sensing and Modeling of Surface Properties 9-11 June 2009, Météo France, Toulouse, France Soil moisture impact on refectance of bare soils in the optical domain [0.4 15 µm] A. Lesaignoux

More information

1. Determine the Zero-Force Members in the plane truss.

1. Determine the Zero-Force Members in the plane truss. 1. Determine the Zero-orce Members in the plane truss. 1 . Determine the force in each member of the loaded truss. Use the Method of Joints. 3. Determine the force in member GM by the Method of Section.

More information

[ ' National Tedocal Information Seract. yäiiiiiliiiyiiiiuilii! PB

[ ' National Tedocal Information Seract. yäiiiiiliiiyiiiiuilii! PB I: p?aiif!!i!? p *!!r?s?!i; n yäiiiiiliiiyiiiiuilii! PB93-37784 Extended Range Modified Gradient Technique for Profile Inversion - Technische Univ. Delft (Netherlands) gpr^s«tor p-^uc leieasal 992 V [

More information

Total Internal Reflection & Metal Mirrors

Total Internal Reflection & Metal Mirrors Phys 531 Lecture 7 15 September 2005 Total Internal Reflection & Metal Mirrors Last time, derived Fresnel relations Give amplitude of reflected, transmitted waves at boundary Focused on simple boundaries:

More information

Week 2: First Order Semi-Linear PDEs

Week 2: First Order Semi-Linear PDEs Week 2: First Order Semi-Linear PDEs Introction We want to find a formal solution to the first order semilinear PDEs of the form a(, y)u + b(, y)u y = c(, y, u). Using a change of variables corresponding

More information

6 Lowercase Letter a Number Puzzles

6 Lowercase Letter a Number Puzzles 1 2 3 4 5 6 7 8 9 10 10 20 30 40 50 60 70 80 90 100 6 Lowercase Letter a Nuber Puzzles 1 2 3 4 5 6 7 8 9 10 10 9 8 7 6 5 4 3 2 1 10 20 30 40 50 60 70 80 90 100 1 2 3 4 5 6 7 8 9 10 10 9 8 7 6 5 4 3 2 1

More information

MATHEMATICS FOR ENGINEERS & SCIENTISTS 23

MATHEMATICS FOR ENGINEERS & SCIENTISTS 23 MATHEMATICS FOR ENGINEERS & SCIENTISTS 3.5. Second order linear O.D.E.s: non-homogeneous case.. We ll now consider non-homogeneous second order linear O.D.E.s. These are of the form a + by + c rx) for

More information

Method of Separation of Variables

Method of Separation of Variables MODUE 5: HEAT EQUATION 11 ecture 3 Method of Separation of Variables Separation of variables is one of the oldest technique for solving initial-boundary value problems (IBVP) and applies to problems, where

More information

THIS PAGE DECLASSIFIED IAW EO IRIS u blic Record. Key I fo mation. Ma n: AIR MATERIEL COMM ND. Adm ni trative Mar ings.

THIS PAGE DECLASSIFIED IAW EO IRIS u blic Record. Key I fo mation. Ma n: AIR MATERIEL COMM ND. Adm ni trative Mar ings. T H S PA G E D E CLA SSFED AW E O 2958 RS u blc Recod Key fo maon Ma n AR MATEREL COMM ND D cumen Type Call N u b e 03 V 7 Rcvd Rel 98 / 0 ndexe D 38 Eneed Dae RS l umbe 0 0 4 2 3 5 6 C D QC d Dac A cesson

More information

fur \ \,,^N/ D7,,)d.s) 7. The champion and Runner up of the previous year shall be allowed to play directly in final Zone.

fur \ \,,^N/ D7,,)d.s) 7. The champion and Runner up of the previous year shall be allowed to play directly in final Zone. OUL O GR SODRY DUTO, ODS,RT,SMTUR,USWR.l ntuctin f cnuct f Kbi ( y/gil)tunent f 2L-Lg t. 2.. 4.. 6. Mtche hll be lye e K ule f ene f tie t tie Dutin f ech tch hll be - +0 (Rece)+ = M The ticint f ech Te

More information

Chapter 1 - The Nature of Light

Chapter 1 - The Nature of Light David J. Starling Penn State Hazleton PHYS 214 Electromagnetic radiation comes in many forms, differing only in wavelength, frequency or energy. Electromagnetic radiation comes in many forms, differing

More information

I cu n y li in Wal wi m hu n Mik an t o da t Bri an Si n. We al ha a c o k do na Di g.

I cu n y li in Wal wi m hu n Mik an t o da t Bri an Si n. We al ha a c o k do na Di g. Aug 25, 2018 De r Fam e, Wel to 7t - ra at Dun Lak! My na is Mr. Van Wag an I te 7t g a la ge ar, an so s u s. Whi t i wi be m 13t ye of te n, it is m 1s ye D S an Cal i Com t Sc o s. I am so ha y an ex

More information

ECE-314 Fall 2012 Review Questions

ECE-314 Fall 2012 Review Questions ECE-34 Fall 0 Review Quesios. A liear ime-ivaria sysem has he ipu-oupu characerisics show i he firs row of he diagram below. Deermie he oupu for he ipu show o he secod row of he diagram. Jusify your aswer.

More information

ECE-342 Test 3: Nov 30, :00-8:00, Closed Book. Name : Solution

ECE-342 Test 3: Nov 30, :00-8:00, Closed Book. Name : Solution ECE-342 Test 3: Nov 30, 2010 6:00-8:00, Closed Book Name : Solution All solutions must provide units as appropriate. Unless otherwise stated, assume T = 300 K. 1. (25 pts) Consider the amplifier shown

More information

Laser-Plasma Interactions

Laser-Plasma Interactions PFC/JA-9-24 Bandwidth of Scattered Radiation in Laser-Plasma Interactions C. C. Chow, A. K. Ram, and A. Bers June 199 Plasma Fusion Center Massachusetts Institute of Technology Cambridge, MA 2139 USA Submitted

More information

Central Results from Newton s Principia Mathematica. I. The Body of Least Resistance

Central Results from Newton s Principia Mathematica. I. The Body of Least Resistance Central Results from Newton s Principia Mathematica I. The Body of Least Resistance Josef Bemelmans Institute for Mathematics, RWTH Aachen February 28 th, 2018 Outline Introduction Proposition XXXIV: The

More information

ELEMENTARY LINEAR ALGEBRA

ELEMENTARY LINEAR ALGEBRA ELEMENTARY LINEAR ALGEBRA K R MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Second Online Version, December 998 Comments to the author at krm@mathsuqeduau All contents copyright c 99 Keith

More information

Differential Equations Practice: 2nd Order Linear: Nonhomogeneous Equations: Undetermined Coefficients Page 1

Differential Equations Practice: 2nd Order Linear: Nonhomogeneous Equations: Undetermined Coefficients Page 1 Differential Equations Practice: 2nd Order Linear: Nonhomogeneous Equations: Undetermined Coefficients Page 1 Questions Example (3.5.3) Find a general solution of the differential equation y 2y 3y = 3te

More information

The Method of Undetermined Coefficients.

The Method of Undetermined Coefficients. The Method of Undetermined Coefficients. James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University May 24, 2017 Outline 1 Annihilators 2 Finding The

More information

Akira Ishimaru, Sermsak Jaruwatanadilok and Yasuo Kuga

Akira Ishimaru, Sermsak Jaruwatanadilok and Yasuo Kuga INSTITUTE OF PHYSICS PUBLISHING Waves Random Media 4 (4) 499 5 WAVES IN RANDOMMEDIA PII: S959-774(4)789- Multiple scattering effects on the radar cross section (RCS) of objects in a random medium including

More information

Presented at the COMSOL Conference 2009 Milan. Analysis of Electromagnetic Propagation for Evaluating the

Presented at the COMSOL Conference 2009 Milan. Analysis of Electromagnetic Propagation for Evaluating the Presented at the COMSOL Conference 2009 Milan Analysis of Electromagnetic Propagation for Evaluating the Dimensions of a Large Lossy Medium A. Pellegrini, FCosta F. 14-16 October 2009 Outline Introduction

More information

THE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE

THE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE THE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE KATIE L. MAY AND MELISSA A. MITCHELL Abstract. We show how to identify the minima path network connecting three fixed points on

More information

0.1 Problems to solve

0.1 Problems to solve 0.1 Problems to solve Homework Set No. NEEP 547 Due September 0, 013 DLH Nonlinear Eqs. reducible to first order: 1. 5pts) Find the general solution to the differential equation: y = [ 1 + y ) ] 3/. 5pts)

More information

IMPACT OF CLIMATE CHANGE ON AGRICULTURAL PRODUCTIVITY AND FOOD SECURITY Khalid Abdul Rahim. A World Leader in New Tropical Agriculture

IMPACT OF CLIMATE CHANGE ON AGRICULTURAL PRODUCTIVITY AND FOOD SECURITY Khalid Abdul Rahim. A World Leader in New Tropical Agriculture IMPACT OF CLIMATE CHANGE ON AGRICULTURAL PRODUCTIVITY AND FOOD SECURITY Khalid Abdul Rahim A World Leader in New Tropical Agriculture IMPACT OF CLIMATE CHANGE ON AGRICULTURAL PRODUCTIVITY AND FOOD SECURITY

More information

Please note that not all pages are included. This is purposely done in order to protect our property and the work of our esteemed composers.

Please note that not all pages are included. This is purposely done in order to protect our property and the work of our esteemed composers. Please note that not all ages ae included This is uosely done in ode to otect ou oety and the wok of ou esteemed comoses f you would like to see this wok in its entiety, lease ode online o call us at 8006472117

More information

PROBLEM 3 AN INVESTIGATION OF DIE DESIGNS FOR TUBE DRAWING. This problan was suggested to the Mathemati os+lrr-lndus tr-y Study Group by

PROBLEM 3 AN INVESTIGATION OF DIE DESIGNS FOR TUBE DRAWING. This problan was suggested to the Mathemati os+lrr-lndus tr-y Study Group by 28 PROBLEM 3 AN INVESTIGATION OF DIE DESIGNS FOR TUBE DRAWING 1. OUTLINE OF PROBLEM This problan was suggested to the Mathemati os+lrr-lndus tr-y Study Group by Metal Manufacturers who produce, among other

More information

Four Ports. 1 ENGN4545/ENGN6545: Radiofrequency Engineering L#16

Four Ports. 1 ENGN4545/ENGN6545: Radiofrequency Engineering L#16 Four Ports Theorems of four port networks 180 o and 90 o Hybrids Directional couplers Transmission line transformers, Four port striplines Rat-races, Wilkinsons, Magic-T s Circulators, Diplexers and Filter

More information

MH CET 2018 (QUESTION WITH ANSWER)

MH CET 2018 (QUESTION WITH ANSWER) ( P C M ) MH CET 8 (QUESTION WITH ANSWER). /.sec () + log () - log (3) + log () Ans. () - log MATHS () 3 c + c C C A cos + cos c + cosc + + cosa ( + cosc ) + + cosa c c ( + + ) c / / I tn - in sec - in

More information

1. The unit vector perpendicular to both the lines. Ans:, (2)

1. The unit vector perpendicular to both the lines. Ans:, (2) 1. The unit vector perpendicular to both the lines x 1 y 2 z 1 x 2 y 2 z 3 and 3 1 2 1 2 3 i 7j 7k i 7j 5k 99 5 3 1) 2) i 7j 5k 7i 7j k 3) 4) 5 3 99 i 7j 5k Ans:, (2) 5 3 is Solution: Consider i j k a

More information