Algebraic Methods in Plane Geometry

Size: px
Start display at page:

Download "Algebraic Methods in Plane Geometry"

Transcription

1 Algebraic Methods in Plane Geometry 1. The Use of Conic Sections Shailesh A Shirali Shailesh Shirali heads a Community Mathematics Center at Rishi Valley School (KFI). He has a deep interest in teaching and writing about mathematics at the high school/post school levels, with particular emphasis on problem solving and the historical aspects of the subject. Written with affection and respect for Professor A R Rao of Ahmedabad, mathematician, teacher and a continuing source of inspiration to a vast number of students, on the occasion of his one hundredth birthday. May there be many more! Keywords. Conics, family of curves, Pascal s theorem, homogeneous coordinates, Butterfly theorem, abelian group, associativity of addition, group law. `R id e r s ' in g e o m e tr y a r e a lw a y s a p le a s u r e t o t a c k le, a n d t h is p le a s u r e is d o u b le d w h e n o n e n d s c o n n e c t io n s b e tw e e n p la n e g e o m e t r y a n d a lg e b r a. T h is t h r e e -p a r t a r t ic le is a b o u t s u c h c o n n e c t io n s. I n P a r t s 1 a n d 2, w e e x p lo r e s o m e c o n n e c t io n s b e t w e e n p la n e g e o m e t r y a n d t h e a l- g e b r a o f c o n ic s a n d c u b ic s ; in P a r t 1 w e g iv e a lg e b r a ic p r o o fs o f r e s u lt s s u c h a s P a s c a l's T h e - o r e m a n d t h e B u t t e r y T h e o r e m, a n d in P a r t 2 w e s t u d y s o m e g r o u p t h e o r e t ic a n d n u m b e r t h e o - r e t ic a s p e c ts o f c u b ic c u r v e s. In P a r t 3 w e lo o k a t t h e r o le o f m a p p in g s a n d t r a n s fo r m a t io n g r o u p s in p la n e g e o m e t r y. 1. P a r a b o la in a T r ia n g le W e rst re ca ll tw o re su lts fro m th e g eo m e try o f th e p a ra b o la. L e t P d e n o te a p a ra b o la w ith fo c u s F a n d d ire c trix `. F o r a n y p o in t P 2 P, le t t P d e n o te th e ta n g en t to P a t P. (i) T h e im a g e o f F u n d e r re e ctio n in a n y o f th e ta n g e n ts t P lie s o n th e d ire c trix `. (S e e F ig u re 1 a ). C o n v erse ly, if th e im a g e o f F u n d e r re e c tio n in a lin e m lie s o n `, th e n m is ta n g e n t to P. (T h e c o lle c tio n o f a ll su ch lin e s m e n v e lo p e s th e p a ra b o la in a v isu a lly v ery a ttra c tiv e w a y, a s c a n b e sh o w n u sin g p a p er fo ld in g.) (ii) If A ; B ; C a re th re e d istin c t p o in ts o n P, th e n th e circ u m c ircle o f th e tria n g le P Q R w h o se sid e s lie o n th e ta n g en ts t A ; t B ; t C, re sp e c tiv ely, p a sse s th ro u g h th e fo c u s F. (S e e F ig u re 1 b ). 916 RESONANCE October 2008

2 (a) (b) S o m e re a d e rs m a y re co g n iz e th a t th e se tw o re su lts co m e to g eth e r in th e W a lla ce{ S im so n th eo rem : \ F o r a n y tria n g le th e fe e t o f th e n o rm a ls fro m a p o in t o n its c irc u m - circ le to th e th re e sid es o f th e tria n g le lie in a stra ig h t lin e ". T h e re su lt b e lo w, w h o se so u rce is a p ro b le m fro m th e p ro b le m so lv in g m a g a zin e C ru x M a th em a tico ru m, b rin g s th ese tw o resu lts in a p retty w a y. T h e o r e m 1. In tria n gle A B C let the feet o f the a ltitu d es fro m A ; B ; C be D ; E ; F, resp ectiv ely. L e t à E! F c u t Ã! A D in K, let L be th e m id po in t o f K D, a n d let th e n o r- m a l to A D a t L cu t à A! C in Q, a n d à A! B in R. T h en th e po in ts A ; R ; D ; Q a re co n cy clic. (S e e F igu re 2.) P roo f. C o n sid e r th e p a ra b o la P w ith fo cu s D, a n d d i- rec trix à E! F (F igu re 3 ). O b se rv e th a t: Figure 1. For any triangle the feet of the normals from a point on its circumcircle to the three sides of the triangle lie in a straight line. Figure 2. RESONANCE October

3 Figure 3. ² à Q! R is ta n ge n t to P. T h is is b ec a u se th e im a g e o f D u n d e r re e ctio n in à Q! R lie s o n th e d ire ctrix à E! F. (T h e im a g e is K.) ² à A! B is ta n g en t to P. T h is is b e c a u se th e im a g e o f D u n d er re ec tio n in à A! B lie s o n th e d irec trix. In tu rn th is is tru e b ec a u se o f th e k n o w n p ro p e rty th a t th e a ltitu d e s A D, B E, C F b ise c t th e a n g les o f th e o rth ic tria n g le 4 D E F. H en c e, F E ; F D m a k e e q u a l a n g le s w ith à A! B, im p ly in g th a t th e im a g e o f D u n d e r re ec tio n in à A! B lies o n à E! F, a s cla im ed. It is a known ² à A! C is ta n gen t to P. T h is is so b ec a u se th e im a g e property of a triangle o f D u n d er re e ctio n in à A! C lies o n th e d ire ctrix (a s formed by three a b o v e). tangents to a ² T h e circu m circle o f 4 A R Q pa sses th ro u gh D. F o r, parabola that its th e ta n g e n ts à R! Q, à A! B, à A! C a re th e sid es o f 4 A R Q, circumcircle passes a n d it is a k n o w n p ro p erty o f a tria n g le fo rm e d b y through the focus of th re e ta n g en ts to a p a ra b o la th a t its c irc u m circ le the parabola. p a sse s th ro u g h th e fo c u s o f th e p a ra b o la. 918 RESONANCE October 2008

4 N o te th a t sin c e à R! Q is p a ra llel to à B! C, th e circu m circles o f 4 A R Q a n d 4 A B C w ill b e ta n g en t to e a ch o th e r a t A, a s F igu re 3 sh o w s. 2. F a m ilie s o f C u r v e s W e n o w sta te a n e x tre m ely u se fu l re su lt c o n c e rn in g fa m - ilies o f cu rves w ith a given d egree. T h e e n tire d iscu ssio n is w ith re fere n c e to a x e d re cta n g u la r c o o rd in a te sy s- tem o n a g iv e n p la n e. C o n sid e r th e fa m ily L o f a ll p o ssib le stra ig h t lin e s. T h e g e n e ra l eq u a tio n o f a stra ig h t lin e is a x + b y + c = 0 (w ith a ; b n o t b o th ze ro ). T h ere a re th re e c o e ± cie n ts in th is eq u a tio n, b u t if w e m u ltip ly a ll o f th em b y a n o n -z e ro co n sta n t w e g e t th e sa m e lin e. H en c e, L h a s `tw o d e g re e s o f free d o m ' (to b o rro w a n e x p ressio n fro m p h y sic s); o r p h ra sed o th e rw ise, L is a tw o p a ra m ete r fa m ily, a n d tw o d istin c t p o in ts a re n e e d ed to x a stra ig h t lin e. If w e x o n e p o in t a n d a llo w th e o th er o n e to v a ry, th e n w e g e t th e o n e p a ra m e ter fa m ily o f a ll lin e s p a ssin g th ro u g h a p o in t, a lso c a lle d a `p e n cil o f lin e s'. T h e te rm `p en cil' m a y b e u n d e rsto o d fro m F igu re 4 a. N e x t, co n sid er th e fa m ily F o f a ll p o ssib le sec o n d d eg ree cu rv e s, i.e., a ll p o ssib le c u rv e s o f th e ty p e p (x ; y ) = 0 ; w h e re p (x ; y ) = a x 2 + by 2 + c x y + d x + e y + f : O b se rv e th a t p h a s = 6 p a ra m e te rs. If w e m u ltip ly p b y a n o n -ze ro co n sta n t, th e c u rv e re m a in s Figure 4. (a) (b) RESONANCE October

5 Five points in general position are needed to fix a second degree curve. th e sa m e, so F is a v e -p a ra m e te r fa m ily. T h e g eo - m e trica l im p lica tio n o f th is is th a t v e p o in ts in ge n era l po sitio n a re n eed ed to x a seco n d d egree cu rve. (In c o n - tra st, th e fa m ily o f c ircles h a s th ree d eg re es o f fre ed o m, a n d th re e p o in ts in g e n e ra l p o sitio n a re n e ed ed to x a circ le. In b o th c a ses, `g en e ra l p o sitio n ' m e a n s th a t n o th re e p o in ts lie in a stra ig h t lin e.) If w e x fo u r p o in ts a n d a llo w th e fth o n e to v a ry, w e g et a o n e p a ra m ete r fa m ily (a `p e n c il o f c o n ic s'), a s th e sk etch in F ig u re 4 b sh o w s. A sim p le b u t e x trem e ly u se fu l co ro lla ry to th e a b o v e o b - se rv a tio n is th e fo llo w in g : If C 1 a n d C 2 a re tw o seco n d d e- gree cu rves pa ssin g th ro u gh fo u r given po in ts A ; B ; C ; D, w ith equ a tio n s p 1 (x ; y ) = 0 a n d p 2 (x ; y ) = 0, re spectively, th en th e equ a tio n o f a n y o th er seco n d d egree cu rve C 3 pa ssin g th ro u gh A ; B ; C ; D m a y be w ritten in th e fo rm 1 p 1 (x ; y ) + 2 p 2 (x ; y ) = 0, w h ere 1 ; 2 a re rea l co n - sta n ts, n o t bo th zero. S im ila r sta te m en ts m a y b e m a d e a b o u t th e p e n cil o f lin es p a ssin g th ro u g h a x e d p o in t, o r a b o u t th e fa m ily o f c irc le s p a ssin g th ro u g h tw o g iv e n p o in ts. (T h is p rin cip le e x te n d s to cu b ic s a s w e ll, b u t w e sh a ll stu d y th is o n ly in P a rt 2.) 3. P a s c a l's H e x a g r a m T h e o r e m W e n o w p ro v e a fa m o u s a n d im p o rta n t th eo rem a b o u t th e co n ic sec tio n s, d u e to B la ise P a sc a l ( { ). T h e o r e m 2 (P a sc a l). T h e o p po site sid es o f a h e xa g o n in scribed in a co n ic in tersect in th ree co llin ea r po in ts. (S e e F ig u re 5.) T h a t is, if six d istin c t p o in ts A ; B ; C ; D ; E ; F lie o n a co n ic C, th e n th e p o in ts o f in te rse ctio n P = Ã A! B \ Ã D! E, Q = Ã B! C \ Ã E! F, R = Ã C! D \ Ã F! A lie in a stra ig h t lin e. It is n o t k n o w n h o w P a sc a l p ro v e d th e th eo re m, o r h o w h e h it u p o n it (w h ich h e d id a t th e a g e o f six te en ). A b o o k h e w a s w ritin g o n th e co n ic sec tio n s circu la te d fo r so m e y ea rs a m o n g th e p ro m in e n t m a th e m a tic ia n s o f E u - ro p e, in d ra ft co p y, a n d th en w a s lo st to m a n k in d. 920 RESONANCE October 2008

6 P a sc a l's th eo rem is a th eo rem o f p ro je ctiv e g e o m e try : it a llo w s th e p o in ts P ; Q ; R to lie o n th e `lin e a t in n ity '. T h u s, if à A! B k à D! E, th en P is `a p o in t a t in n ity '. In th is ca se th e th e o re m im p lie s th a t à Q! R k à A! B. If it h a p p e n s th a t à A! B k à D! E a s w e ll a s à B! C k à E! F, th en th e th eo rem a sserts th a t à C! D k à F! A. (N o w a ll th re e o f P ; Q ; R lie o n th e lin e a t in n ity.) A n im p o rta n t co n se q u e n c e o f th e p ro jec tiv e v ie w p o in t is th a t th e lin e a t in n ity d o e s n o t h a v e a n y sp e cia l sta tu s; it is trea te d o n p a r w ith e v e ry o th e r lin e. S o in th e p ro je ctiv e p ro o f, it is o f n o c o n se q u en ce if so m e p a irs o f lin e s a re p a ra lle l to e a ch o th er; th e w o rd in g o f th e p ro o f re m a in s e x a c tly th e sa m e. T o im p lem e n t th is a p p ro a ch, w e u se p ro jective coo rd in a tes, in w h ich p o in ts a re d e n o te d u sin g trip les [x ; y ; z ] o f re a l n u m b e rs. H ere a re th e b a sic ru le s g o v e rn in g th e se trip le s: (i) x ; y ; z a re n o t a ll z e ro ; (ii) [k x ; k y ; k z ] d e n o te s th e sa m e p o in t a s [x ; y ; z ] fo r a n y rea l n u m b er k 6= 0. T h e u n d e rsta n d in g is th a t if z 6= 0, th e n [x ; y ; z ] co rresp o n d s to th e p o in t w ith ca rtesia n c o o rd in a te s (x = z ; y = z ), a n d if z = 0 th e n [x ; y ; z ] lie s o n th e lin e a t in n ity. T h e lin e w ith c a rte sia n eq u a tio n x + y = 1 a cq u ires th e e q u a tio n x + y z = 0 in th is sy ste m, w h ile th e c ircle w ith c a rte sia n e q u a tio n Figure 5. An important consequence of the projective viewpoint is that the line at infinity does not have any special status; it is treated on par with every other line. RESONANCE October

7 x 2 + y 2 x y = 1 a c q u ire s th e e q u a tio n x 2 + y 2 x z y z z 2 = 0. N o te th a t th e se e q u a tio n s a re h o m og en eo u s. T h e lin e a t in n ity h a s th e eq u a tio n z = 0. P roo f o f P a sca l's T h eo rem. L e t L A B (x ; y ; z ) = 0, L B C (x ; y ; z ) = 0, L C D (x ; y ; z ) = 0, : : : b e th e e q u a tio n s o f à A! B, à B! C, à C! D : : :, re sp e c - tiv ely, w h e re L A B, L B C, L C D, : : : a re lin e a r e x p re ssio n s in x ; y ; z. L e t f (x ; y ; z ) = 0 b e th e e q u a tio n o f th e c o n ic, w h e re f is a p o ly n o m ia l in x ; y ; z o f d e g re e 2. T h e v a rio u s L 's a n d f a re h o m o g en e o u s e x p ressio n s. S in ce C p a sses th ro u g h A ; B ; C ; D, a n d so d o th e tw o p a ir-o f-stra ig h t-lin e s c o n ic s à A! B [ à C! D a n d à A! D [ à B! C (see F igu re 6 ), th e re ex ist c o n sta n ts ; 0 su ch th a t f = L A B L C D + 0 L A D L B C : (1 ) S im ila rly, sin ce C p a sse s th ro u g h A ; E ; F ; D, a n d so d o th e tw o c o n ics à A! D [ à E! F a n d à D! E [ à A! F, th e re e x ist c o n - sta n ts ; 0 su ch th a t f = L A D L E F + 0L D E L A F : (2 ) F ro m (1 ) a n d (2 ) w e g e t L A B L C D + 0 L A D L B C = L A D L E F + 0L D E L A F ; ) L A D ( 0 L B C L E F ) = 0L D E L A F L A B L C D : (3 ) Figure RESONANCE October 2008

8 L e t p o ly n o m ia ls g ; h b e d e n e d a s fo llo w s: ½ g = 0L D E L A F L A B L C D ; h = 0 L B C L E F : (4 ) S in ce it c a n n o t h a p p e n th a t h is id e n tica lly 0 (th is w o u ld m a k e à B! C c o in c id e w ith à A! D, w h ich is n o t a llo w e d b y th e h y p o th e se s o f th e th e o re m ), it m u st b e th a t h h a s d eg ree 1. T h e n th e e q u a tio n h = 0 rep re se n ts a stra ig h t lin e. N o w o b serv e th a t: ² P 2 à A! B a n d P 2 à D! E, so g (P ) = 0. S in c e P 62 à A! D, it fo llo w s th a t h (P ) = 0. ² R 2 à C! D a n d R 2 à A! F, so g (R ) = 0. S in ce R 62 à A! D, it fo llo w s th a t h (R ) = 0. ² Q 2 à B! C a n d Q 2 à E! F, so h (Q ) = 0. When C is a pair! of straight lines we get the result known as Pappus s theorem. S o P ; Q ; R lie o n th e lin e 0 L B C L E F = 0, a n d a re th u s c o llin e a r. R e m a r k s ² T h e p ro o f d o e s n o t c o n sid e r sep a ra te ly th e ca se s o f p a ra lle lism. (It is n o t req u ired, a s p er th e re m a rk s m a d e a b o v e.) ² P a sca l's th e o re m is e x trem e ly w id e in its sc o p e : th e h e x a g o n A B C D E F m a y b e n o n -c o n v e x a n d / o r selfin terse c tin g, a n d th e c o n ic C itse lf m a y b e a n y se c - o n d d eg ree lo cu s. W h e n C is a p a ir o f stra ig h t lin e s w e g et th e re su lt k n o w n a s P a p p u s's th e o re m. ² W e m a y ev en a llo w so m e p a irs o f p o in ts to co in c id e; in th is ca se w e g et `lim itin g c a se s' o f th e th eo rem b y re p la cin g `lin e s' b y `ta n g e n ts' a s n e e d e d. F o r e x a m p le if w e le t A co in c id e w ith B, w e g e t th e fo l- lo w in g sta tem e n t, in w h ich t X d en o tes th e ta n g en t to th e co n ic a t a n y p o in t X o n it: If B ; C ; D ; E ; F a re ve d istin ct po in ts o n a co n ic C, th en th e po in ts P = t B \ à D! E, Q = à B! C \ à E! F, R = à C! D \ à E! F lie in a stra igh t lin e. RESONANCE October

9 If ABCDEF is a hexagon formed by six lines that are all tangent to a conic, then the lines AD, BE, CF concur. ² A sim ila r p ro o f m a y b e d ev ised fo r B ria n ch o n 's th e - o rem : If A B C D E F is a h exago n fo rm ed by six lin es th a t a re a ll ta n g en t to a co n ic, th e n th e lin e s à A! D, Ã! B E, à C! F co n c u r. 4. T h e B u tt e r y T h e o r e m T h e b u tte r y th e o re m rst a p p e a re d a s a p ro b lem in th e e a rly 's, a n d o n e o f th e ea rly p ro o fs is d u e to W G H o rn er w h o is b e tter k n o w n fo r a n a lg o rith m in p o ly n o m ia l a rith m e tic. T h e o r e m 3 (B u tter y T h eo rem ). L et P Q be a ch o rd o f circle K, a n d let M be its m id po in t. T h ro u gh M tw o o th er ch o rd s A B a n d C D a re d ra w n. L et A D a n d B C cu t Ã! P Q in E a n d F, respectively. T h en M is th e m id po in t o f E F. (S e e F igu re 7.) A la rg e n u m b e r o f e le g a n t p ro o fs o f th e B u tter y T h eo - rem h a v e a p p e a re d o v e r th e y e a rs, in clu d in g m a n y `p u re g e o m e try ' p ro o fs; b u t th e fo llo w in g p ro o f is p a rtic u la rly ch a rm in g, fo r it sim u lta n eo u sly ca sts th e th eo rem in a m o re g en e ra l se ttin g a n d m a k e s it e a sy to fo rm u la te sp e - cia l c a se s o f in te re st. It is b a se d y et a g a in o n th e a sse r- tio n s m a d e in S e c tio n 2. Figure RESONANCE October 2008

10 T h e o r e m 4 (B u tter y T h eo rem fo r C o n ic s). L et P Q be a ch o rd o f a co n ic K, a n d let M be its m id po in t. T h ro u gh M let tw o o th er ch o rd s A B a n d C D be d ra w n. L et A D a n d B C cu t à P! Q in E a n d F, respectively. T h en M is th e m id po in t o f E F. (S ee F igu re 8.) P roo f. W e u se a c a rte sia n se ttin g, w ith Ã! P Q a s th e x -a x is, a n d M a s th e o rig in. L e t K h a v e eq u a tio n p (x ; y ) = 0, w h e re p (x ; y ) = a x 2 + b y 2 + c x y + d x + e y + f. T h e in te rse ctio n s o f K w ith th e x -a x is a re fo u n d b y so lv in g th e eq u a tio n s p (x ; y ) = 0, y = 0, i.e., a x 2 + d x + f = 0 : (5 ) S in ce M is th e m id p o in t o f P Q, th e su m o f th e ro o ts o f (5 ) is 0, w h ich im p lie s th a t d = 0. H e n ce, th e coe ± cien t o f th e x -term in p (x ; y ) is zero. T h is a ssertio n a lso h o ld s fo r th e lin e -p a ir co n ic K 0 = Ã! A B [ à C! D, b ec a u se th e lin e s à A! B a n d à C! D p a ss th ro u g h th e o rig in. T h a t is, if th e equ a tio n o f K 0 is q (x ; y ) = 0, th en th e coe± cien t o f th e x -term in q (x ; y ) is zero. S in ce K a n d K 0 sh a re th e fo u r p o in ts A ; B ; C ; D, th e sa m e sta tem en t is tru e fo r a n y co n ic th a t pa sses th ro u gh A ; B ; C ; D. T h is is b ec a u se th e e q u a tio n o f a n y su ch co n ic is o f th e fo rm r p (x ; y ) + s q (x ; y ) = 0, w h e re r ; s a re re a l n u m b ers. Figure 8. RESONANCE October

11 A surprising consequence of Pascal s theorem is that it allows us to define a group on the points of any nondegenerate conic. GENERAL ARTICLE In p a rticu la r it is tru e fo r th e lin e-p a ir c o n ic K 00 = à A! C [ Ã! B D. H en c e, th e su m o f th e ro o ts o f th e in te rse c tio n s o f K 00 w ith th e x -a x is is 0. T h a t is, M is th e m id p o in t o f E F. H e re a re tw o ty p ic a l `sp ec ia l c a se s' w h o se p ro o fs w e le a v e to th e re a d er: T h e o r e m 5. L et A B C D be a cy clic qu a d rila tera l w h o se circu m circle K h a s A C a s a d ia m eter, a n d O a s its cen - ter. L et th e ta n gen t to K a t A m eet à B! D a t P ; let à P! O m eet à C! B in E, a n d à C! D in F, respectively. T h en O is th e m id po in t o f E F. (S ee F igu re 9.) T h e o r e m 6. L et A B, C D be segm en ts in tersectin g a t a po in t P, a n d let Q a n d R be po in ts o n A B a n d C D, respectively, su ch th a t A P = Q B, a n d C P = R D. L et Ã! Q R m eet A D in U, a n d B C in V. T h en U R = Q V. (S ee F igu re 1 0.) 5. G r o u p o n a C o n ic Figure 9 (left). Figure 10 (right). In c lo sin g w e p o in t o u t a ra th e r su rp risin g c o n se q u en ce o f P a sc a l's th e o re m : it a llo w s u s to d e n e a g ro u p o n th e p o in ts o f a n y n o n -d e g e n e ra te c o n ic. L et N b e a n y x e d p o in t o f su ch a co n ic K ; th is w ill se rv e a s th e n e u tra l p o in t, i.e., th e id e n tity e lem e n t o f th e g ro u p. W e d e n e 926 RESONANCE October 2008

12 (a) (b) th e b in a ry o p era tio n th u s: if P ; Q a re p o in ts o n K, w e d ra w th ro u g h N a lin e p a ra lle l to à P! Q ; th en P Q is th e p o in t w h ere th is lin e in te rse cts K a g a in. T h e co n stru c tio n is a s d e p icte d in F igu re 1 1 a. Figure 11. It is ea sy to ch e ck th e fo llo w in g a ssertio n s: ² T h e o p era tio n is w ell d e n e d. F o r, th e lin e th ro u g h N p a ra llel to à P! Q w ill in te rse ct K a se co n d tim e, sin c e K is a sec o n d d e g re e cu rv e. ² P Q = Q P fo r a ll p a irs o f p o in ts P ; Q 2 K ; so is c o m m u ta tiv e. ² If P = Q, th e n Ã! P Q is ta n g en t to K a t P. S o to n d P P, w e d ra w a lin e th ro u g h N p a ra lle l to th e ta n g e n t t P to K a t P ; th e n its se c o n d p o in t o f in terse c tio n w ith K is P P. S ee F igu re 1 1 b. ² W e h a v e N N = N, a n d P N = P fo r a ll P 2 K. ² If th e lin e th ro u g h N p a ra lle l to Ã! P Q is ta n g e n t to K a t N, th e n R = N, so w e w rite P Q = N. T h is e n a b le s u s to n d in v erses. If w e in v o k e P a sca l's th eo rem in th is co n g u ra tio n w e n d fa irly e a sily th a t P (Q R ) = (P Q ) R fo r a n y th re e p o in ts P ; Q ; R o n K. T h is im p lie s th e fo llo w in g resu lt. T h e o r e m 7. T h e pa ir (K ; ) is a n a belia n gro u p, w ith N a s th e id en tity elem en t. RESONANCE October

13 It is in te restin g to c la ssify th e g ro u p s co rresp o n d in g to d i e ren t c o n ic s. H e re a re th e m a in re su lts, w h o se p ro o fs w e le a v e to th e rea d e r: 1. If K is eith e r a n ellip se o r a c ircle, th e n (K ; ) is iso m o rp h ic to th e m u ltip lic a tiv e g ro u p o f c o m p le x n u m b e rs w ith u n it m a g n itu d e, i.e., iso m o rp h ic to th e g ro u p R = 2 ¼ Z. 2. If K is a p a ra b o la, th e n (K ; ) is iso m o rp h ic to th e a d d itiv e g ro u p o f re a l n u m b e rs, (R ; + ). 3. If K is a h y p e rb o la, th e n (K ; ) is iso m o rp h ic to th e m u ltip lic a tiv e g ro u p o f n o n -z ero re a l n u m b e rs, (R? ; ). T h e re a re n u m e ro u s q u estio n s o f in te re st w h ich c a n b e ex p lo re d in th is re g a rd, in c lu d in g so m e w h ich a re o f a n u m b e r-th e o re tic n a tu re. T h e se a rise w h en o n e re g a rd s th e co n ic a s d e n e d o v e r so m e n ite eld ra th er th a n th e e ld o f re a l n u m b ers; e.g., o v e r th e in te g e rs m o d u lo p fo r so m e p rim e p. T h e c o n ic in su ch a ca se is n o t a v isu a liz a b le o b jec t, a n d o n e h a s to stu d y it a lg e b ra ic a lly ra th e r th a n g eo m e trica lly. H e re is a ty p ic a l resu lt: If K is th e co n ic x 2 2 y 2 = 1 o ve r th e eld o f in tege rs m od u lo a p rim e p, th en (K ; ) is a cy clic gro u p ; its o rd er is p 1 if 2 is a qu a d ra tic resid u e m od u lo p, a n d p + 1 if 2 is a qu a d ra tic n o n re sid u e m o d u lo p. H o w ev er w e sh a ll n o t d w ell o n th is to p ic h ere. T h e re a d e r is re ferre d to [4 ] fo r fu rth e r d e ta ils. Suggested Reading Address for Correspondence Shailesh A Shirali Rishi Valley School Rishi Valley Madanapalle, AP, India. shailesh.shirali@gmail.com [1] H S M Coxeter, Introduction to Geometry, 2nd edition, Wiley, [2] H S M Coxeter and S L Greitzer, Geometry Revisited, Math. Assoc. Amer., 1st edition, [3] S L Loney, The Elements of Coordinate Geometry, in two volumes, MacMillan India, [4] Shailesh Shirali, Groups Associated With Conics, The Mathematical Gazette, (to appear February 2009). [5] Eric W Weisstein, Conic Section, From MathWorld A Wolfram Web Resource, RESONANCE October 2008

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

LU N C H IN C LU D E D

LU N C H IN C LU D E D Week 1 M o n d a y J a n u a ry 7 - C o lo u rs o f th e R a in b o w W e w ill b e k ic k in g o ff th e h o lid a y s w ith a d a y fu ll o f c o lo u r! J o in u s fo r a ra n g e o f a rt, s p o rt

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

Model Checking. Automated Verification of Computational Systems

Model Checking. Automated Verification of Computational Systems Model Checking Automated Verification of Computational Systems Madhavan Mukund T h e A C M T u r in g A w a r d fo r 2 0 0 7 w a s a w a r d e d t o C la r k e, E m e r s o n a n d S ifa k is fo r t h

More information

600 Billy Smith Road, Athens, VT

600 Billy Smith Road, Athens, VT 600 Billy Smith Road, Athens, VT Curtis Trousdale, Owner, Broker, Realtor Cell: 802-233-5589 curtis@preferredpropertiesvt.com 2004 Williston Road, South Burlington VT 05403 www.preferredpropertiesvt.com

More information

Class Diagrams. CSC 440/540: Software Engineering Slide #1

Class Diagrams. CSC 440/540: Software Engineering Slide #1 Class Diagrams CSC 440/540: Software Engineering Slide # Topics. Design class diagrams (DCDs) 2. DCD development process 3. Associations and Attributes 4. Dependencies 5. Composition and Constraints 6.

More information

Grain Reserves, Volatility and the WTO

Grain Reserves, Volatility and the WTO Grain Reserves, Volatility and the WTO Sophia Murphy Institute for Agriculture and Trade Policy www.iatp.org Is v o la tility a b a d th in g? De pe n d s o n w h e re yo u s it (pro d uc e r, tra d e

More information

gender mains treaming in Polis h practice

gender mains treaming in Polis h practice gender mains treaming in Polis h practice B E R L IN, 1 9-2 1 T H A P R IL, 2 O O 7 Gender mains treaming at national level Parliament 25 % of women in S ejm (Lower Chamber) 16 % of women in S enat (Upper

More information

EKOLOGIE EN SYSTEMATIEK. T h is p a p e r n o t to be c i t e d w ith o u t p r i o r r e f e r e n c e to th e a u th o r. PRIMARY PRODUCTIVITY.

EKOLOGIE EN SYSTEMATIEK. T h is p a p e r n o t to be c i t e d w ith o u t p r i o r r e f e r e n c e to th e a u th o r. PRIMARY PRODUCTIVITY. EKOLOGIE EN SYSTEMATIEK Ç.I.P.S. MATHEMATICAL MODEL OF THE POLLUTION IN NORT H SEA. TECHNICAL REPORT 1971/O : B i o l. I T h is p a p e r n o t to be c i t e d w ith o u t p r i o r r e f e r e n c e to

More information

STEEL PIPE NIPPLE BLACK AND GALVANIZED

STEEL PIPE NIPPLE BLACK AND GALVANIZED Price Sheet Effective August 09, 2018 Supersedes CWN-218 A Member of The Phoenix Forge Group CapProducts LTD. Phone: 519-482-5000 Fax: 519-482-7728 Toll Free: 800-265-5586 www.capproducts.com www.capitolcamco.com

More information

Form and content. Iowa Research Online. University of Iowa. Ann A Rahim Khan University of Iowa. Theses and Dissertations

Form and content. Iowa Research Online. University of Iowa. Ann A Rahim Khan University of Iowa. Theses and Dissertations University of Iowa Iowa Research Online Theses and Dissertations 1979 Form and content Ann A Rahim Khan University of Iowa Posted with permission of the author. This thesis is available at Iowa Research

More information

MOLINA HEALTHCARE, INC. (Exact name of registrant as specified in its charter)

MOLINA HEALTHCARE, INC. (Exact name of registrant as specified in its charter) UNITED STATES SECURITIES AND EXCHANGE COMMISSION Washington, D.C. 20549 FORM 8-K Current Report Pursuant to Section 13 or 15(d) of the Securities Exchange Act of 1934 Date of Report (Date of earliest event

More information

UNITED STATES SECURITIES AND EXCHANGE COMMISSION Washington, D.C Form 8-K/A (Amendment No. 2)

UNITED STATES SECURITIES AND EXCHANGE COMMISSION Washington, D.C Form 8-K/A (Amendment No. 2) UNITED STATES SECURITIES AND EXCHANGE COMMISSION Washington, D.C. 20549 Form 8-K/A (Amendment No. 2) Current Report Pursuant to Section 13 or 15(d) of the Securities Exchange Act of 1934 Date of Report

More information

TTM TECHNOLOGIES, INC. (Exact Name of Registrant as Specified in Charter)

TTM TECHNOLOGIES, INC. (Exact Name of Registrant as Specified in Charter) Table of Contents UNITED STATES SECURITIES AND EXCHANGE COMMISSION WASHINGTON, DC 20549 FORM 8-K CURRENT REPORT Pursuant to Section 13 or 15(d) of the Securities Exchange Act of 1934 November 15, 2006

More information

B ooks Expans ion on S ciencedirect: 2007:

B ooks Expans ion on S ciencedirect: 2007: B ooks Expans ion on S ciencedirect: 2007: 1 INFORUM, 22-24 May, Prague Piotr Golkiewicz Account Manager Elsevier B.V. Email: p.golkiewicz@elsevier.com Mobile: +48 695 30 60 17 2 Pres entation Overview

More information

UNITED STATES SECURITIES AND EXCHANGE COMMISSION Washington, D.C FORM 8-K

UNITED STATES SECURITIES AND EXCHANGE COMMISSION Washington, D.C FORM 8-K UNITED STATES SECURITIES AND EXCHANGE COMMISSION Washington, D.C. 20549 FORM 8-K CURRENT REPORT Pursuant to Section 13 or 15(d) of the Securities Exchange Act of 1934 Date of Report (Date of earliest event

More information

Distributive Justice, Injustice and Beyond Justice: The Difference from Principle to Reality between Karl Marx and John Rawls

Distributive Justice, Injustice and Beyond Justice: The Difference from Principle to Reality between Karl Marx and John Rawls W CP 2 0 0 8 P ro c e e d in g s V o l.5 0 S o cia l a n d P o litic a l P h ilo s o p h y Distributive Justice, Injustice and Beyond Justice: The Difference from Principle to Reality between Karl Marx

More information

MONTHLY REVIEW. f C r e d i t a n d B u s i n e s s C o n d i t i o n s F E D E R A L R E S E R V E B A N K O F N E W Y O R K MONEY MARKET IN JUNE

MONTHLY REVIEW. f C r e d i t a n d B u s i n e s s C o n d i t i o n s F E D E R A L R E S E R V E B A N K O F N E W Y O R K MONEY MARKET IN JUNE MONTHLY REVIEW O f C r e d i t a n d B u s i n e s s C o n d i t i o n s F E D E R A L R E S E R V E B A N K O F N E W Y O R K V o l u m e 38 J U L Y 1956 No. 7 P re s su re s o n m e m b e r b a n k re

More information

THE BANK OF NEW YORK MELLON CORPORATION (Exact name of registrant as specified in its charter)

THE BANK OF NEW YORK MELLON CORPORATION (Exact name of registrant as specified in its charter) UNITED STATES SECURITIES AND EXCHANGE COMMISSION Washington, D.C. 20549 FORM 8-K CURRENT REPORT Pursuant to Section 13 or 15(d) of the Securities Exchange Act of 1934 Date of Report (Date of earliest event

More information

Functional pottery [slide]

Functional pottery [slide] Functional pottery [slide] by Frank Bevis Fabens A thesis submitted in partial fulfillment of the requirements for the degree of Master of Fine Arts Montana State University Copyright by Frank Bevis Fabens

More information

C o r p o r a t e l i f e i n A n c i e n t I n d i a e x p r e s s e d i t s e l f

C o r p o r a t e l i f e i n A n c i e n t I n d i a e x p r e s s e d i t s e l f C H A P T E R I G E N E S I S A N D GROWTH OF G U IL D S C o r p o r a t e l i f e i n A n c i e n t I n d i a e x p r e s s e d i t s e l f i n a v a r i e t y o f f o r m s - s o c i a l, r e l i g i

More information

REFUGEE AND FORCED MIGRATION STUDIES

REFUGEE AND FORCED MIGRATION STUDIES THE OXFORD HANDBOOK OF REFUGEE AND FORCED MIGRATION STUDIES Edited by ELENA FIDDIAN-QASMIYEH GIL LOESCHER KATY LONG NANDO SIGONA OXFORD UNIVERSITY PRESS C o n t e n t s List o f Abbreviations List o f

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

AGRICULTURE SYLLABUS

AGRICULTURE SYLLABUS Agriculture Forms 1-4.qxp_Layout 1 26/10/2016 12:29 PM Page 1 ZIMBABWE MInISTRY OF PRIMARY AnD SECOnDARY EDUCATIOn AGRICULTURE SYLLABUS FORM 1-4 2015-2022 Curriculum Development and Technical Services,

More information

The Ability C ongress held at the Shoreham Hotel Decem ber 29 to 31, was a reco rd breaker for winter C ongresses.

The Ability C ongress held at the Shoreham Hotel Decem ber 29 to 31, was a reco rd breaker for winter C ongresses. The Ability C ongress held at the Shoreham Hotel Decem ber 29 to 31, was a reco rd breaker for winter C ongresses. Attended by m ore than 3 00 people, all seem ed delighted, with the lectu res and sem

More information

Th e E u r o p e a n M ig r a t io n N e t w o r k ( E M N )

Th e E u r o p e a n M ig r a t io n N e t w o r k ( E M N ) Th e E u r o p e a n M ig r a t io n N e t w o r k ( E M N ) H E.R E T h em at ic W o r k sh o p an d Fin al C o n fer en ce 1 0-1 2 Ju n e, R agu sa, It aly D avid R eisen zein IO M V ien n a Foto: Monika

More information

Computer Games as a Pedagogical Tool in Education. Ken Maher B.Sc. School of Computer Applications, Dublin City University, Glasnevin, Dublin 9.

Computer Games as a Pedagogical Tool in Education. Ken Maher B.Sc. School of Computer Applications, Dublin City University, Glasnevin, Dublin 9. Computer Games as a Pedagogical Tool in Education By Ken Maher B.Sc. School of Computer Applications, Dublin City University, Glasnevin, Dublin 9. / / Supervisor: Dr Micheál O heigeartaigh A Dissertation

More information

ANNUAL MONITORING REPORT 2000

ANNUAL MONITORING REPORT 2000 ANNUAL MONITORING REPORT 2000 NUCLEAR MANAGEMENT COMPANY, LLC POINT BEACH NUCLEAR PLANT January 1, 2000, through December 31, 2000 April 2001 TABLE OF CONTENTS Executive Summary 1 Part A: Effluent Monitoring

More information

UNITED STATES SECURITIES AND EXCHANGE COMMISSION FORM 8-K. Farmer Bros. Co.

UNITED STATES SECURITIES AND EXCHANGE COMMISSION FORM 8-K. Farmer Bros. Co. UNITED STATES SECURITIES AND EXCHANGE COMMISSION Washington, D.C. 20549 FORM 8-K CURRENT REPORT PURSUANT TO SECTION 13 OR 15(d) OF THE SECURITIES EXCHANGE ACT OF 1934 Date of Report (Date of earliest event

More information

TECHNICAL MANUAL OPTIMA PT/ST/VS

TECHNICAL MANUAL OPTIMA PT/ST/VS TECHNICAL MANUAL OPTIMA PT/ST/VS Page 2 NT1789 Rév.A0 TABLE OF CHANGES The information contained in this document only concerns : OPTIMA PT/ST/VS type, MCM 440 PT/OT type, MCM550 ST type. Technical Manual

More information

M a n a g e m e n t o f H y d ra u lic F ra c tu rin g D a ta

M a n a g e m e n t o f H y d ra u lic F ra c tu rin g D a ta M a n a g e m e n t o f H y d ra u lic F ra c tu rin g D a ta M a rc h 2 0 1 5, A n n a F ilip p o v a a n d J e re m y E a d e 1 W h a t is H y d ra u lic F ra c tu rin g? Im a g e : h ttp ://w w w.h

More information

Joh n L a w r e n c e, w ho is on sta ff at S ain t H ill, w r ite s :

Joh n L a w r e n c e, w ho is on sta ff at S ain t H ill, w r ite s : Minor Issue 168 S C I E N T O L O G Y A N D C H I L D R E N T h e r e a r e at p r e s e n t no b o o k s a v a ila b le on th e su b je c t of te a c h in g S c ie n to lo g y to c h ild r e n. A s th

More information

Country Report Government (Part I) Due: November 14, 2017

Country Report Government (Part I) Due: November 14, 2017 Country Report Government (Part I) Due: November 14, 2017 o The government has a total of three sections: government, flag, and national anthem. You will start by researching your government. o Step 1:

More information

heliozoan Zoo flagellated holotrichs peritrichs hypotrichs Euplots, Aspidisca Amoeba Thecamoeba Pleuromonas Bodo, Monosiga

heliozoan Zoo flagellated holotrichs peritrichs hypotrichs Euplots, Aspidisca Amoeba Thecamoeba Pleuromonas Bodo, Monosiga Figures 7 to 16 : brief phenetic classification of microfauna in activated sludge The considered taxonomic hierarchy is : Kingdom: animal Sub kingdom Branch Class Sub class Order Family Genus Sub kingdom

More information

S ca le M o d e l o f th e S o la r Sy ste m

S ca le M o d e l o f th e S o la r Sy ste m N a m e ' D a t e ' S ca le M o d e l o f th e S o la r Sy ste m 6.1 I n t r o d u c t i o n T h e S olar System is large, at least w hen com pared to distances we are fam iliar w ith on a day-to-day basis.

More information

A Study of Attitude Changes of Selected Student- Teachers During the Student-Teaching Experience.

A Study of Attitude Changes of Selected Student- Teachers During the Student-Teaching Experience. Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1973 A Study of Attitude Changes of Selected Student- Teachers During the Student-Teaching Experience.

More information

Comparative Analyses of Teacher Verbal and Nonverbal Behavior in a Traditional and an Openspace

Comparative Analyses of Teacher Verbal and Nonverbal Behavior in a Traditional and an Openspace East Tennessee State University Digital Commons @ East Tennessee State University Electronic Theses and Dissertations June 1975 Comparative Analyses of Teacher Verbal and Nonverbal Behavior in a Traditional

More information

University Microfilms

University Microfilms University Microfilms International * i---------------------------------------------------------- ----------------------- MICROCOPY RESOLUTION TEST CHART N ATIO NAL HI IH l A l l o t ST AN PAR P S II A

More information

UNITED STATES SECURITIES AND EXCHANGE COMMISSION WASHINGTON, D.C FORM 8-K

UNITED STATES SECURITIES AND EXCHANGE COMMISSION WASHINGTON, D.C FORM 8-K Table of Contents UNITED STATES SECURITIES AND EXCHANGE COMMISSION WASHINGTON, D.C. 20549 FORM 8-K CURRENT REPORT PURSUANT TO SECTION 13 OR 15(d) OF THE SECURITIES EXCHANGE ACT OF 1934 DATE OF REPORT (DATE

More information

M I E A T? Y A H 0E 3TE S

M I E A T? Y A H 0E 3TE S M I E A T? Y A H 0E 3TE S Corrgimi c a tod to the- Councl 1 and 1,'ombors ox the League 3/36456712247 p 9 AP t * no 1 Q A L» * O i-» m i. i O JL /» X T T i ttt.' n *7 T-T * n i T n TTi U U jj!.» -! 1 Uj.']

More information

A new ThermicSol product

A new ThermicSol product A new ThermicSol product Double-Faced Thermo-Electric Solar-Panel TD/PV & Solar Tracker & Rotation Device An EU-patent protected product TP4-referens.pdf D o y o u w a n t to c o n v e rt it i n to G re

More information

NORWEGIAN MARITIME DIRECTORATE

NORWEGIAN MARITIME DIRECTORATE PAME Snap shot Analysis NORWEGIAN MARITIME DIRECTORATE PAME Snap Shot Analysis of Maritime Activities in the Arctic Revision No. 01 REPORT NO. 2000-3220 Page 1 PAME Snap shot Analysis Table of Contents

More information

Sub: Filing of Reconciliation of share capital for the quarter ended September 30, 2018

Sub: Filing of Reconciliation of share capital for the quarter ended September 30, 2018 I N D I A Tl F in an c ial H o ld in g s L im ite d (F o rm e rly k n o w n as T u b e In v e s tm e n ts o f In d ia L im ite d ) Dare House, 234, N.S.C. Bose Road, Chennai 600 001, India Tel: 91.44.4217

More information

LSU Historical Dissertations and Theses

LSU Historical Dissertations and Theses Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1976 Infestation of Root Nodules of Soybean by Larvae of the Bean Leaf Beetle, Cerotoma Trifurcata

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

The Effects of Apprehension, Conviction and Incarceration on Crime in New York State

The Effects of Apprehension, Conviction and Incarceration on Crime in New York State City University of New York (CUNY) CUNY Academic Works Dissertations, Theses, and Capstone Projects Graduate Center 1978 The Effects of Apprehension, Conviction and Incarceration on Crime in New York State

More information

McCormick & Company, Incorporated (Exact name of registrant as specified in its charter)

McCormick & Company, Incorporated (Exact name of registrant as specified in its charter) UNITED STATES SECURITIES AND EXCHANGE COMMISSION Washington, D.C. 20549 FORM 8-K CURRENT REPORT Pursuant to Section 13 or 15(d) of the Securities Exchange Act of 1934 Date of Report (Date of earliest event

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

Space and Time in Life and Science

Space and Time in Life and Science Space and Time in Life and Science Vasant Natarajan, V Balakrishnan and N Mukunda S p a c e a n d tim e a r e c o n c e p t s t h a t s e e m t o b e e m - b e d d e d in o u r v e r y c o n s c io u s

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

CHAPTER 6 SUMMARV, m a in FINDIN6S AND C0NCUL5I0NS

CHAPTER 6 SUMMARV, m a in FINDIN6S AND C0NCUL5I0NS CHAPTER 6 SUMMARV, m a in FINDIN6S AND C0NCUL5I0NS 6.1; AFRICA AND SOUTHERN AFRICA Africa was the world's first continent where not only man evolved but also the human civilization. It is the largest continent

More information

R e p u b lic o f th e P h ilip p in e s. R e g io n V II, C e n tra l V isa y a s. C ity o f T a g b ila ran

R e p u b lic o f th e P h ilip p in e s. R e g io n V II, C e n tra l V isa y a s. C ity o f T a g b ila ran R e p u b lic o f th e P h ilip p in e s D E P A R T M E N T O F E D U C A T IO N R e g io n V II, C e n tra l V isa y a s D IV IS IO N O F B O H O L C ity o f T a g b ila ran S e p te m b e r 2 8, 2 0

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

Breakup of weakly bound nuclei and its influence on fusion. Paulo R. S. Gomes Univ. Fed. Fluminense (UFF), Niteroi, Brazil

Breakup of weakly bound nuclei and its influence on fusion. Paulo R. S. Gomes Univ. Fed. Fluminense (UFF), Niteroi, Brazil Breakup of weakly bound nuclei and its influence on fusion Paulo R. S. Gomes Univ. Fed. Fluminense (UFF), Niteroi, Brazil Forum Brasil-JINR Dubna, June, 2015 For a comprehensive review of this subject

More information

R e p u b lic o f th e P h ilip p in e s. C ity o f T a g b ila ran

R e p u b lic o f th e P h ilip p in e s. C ity o f T a g b ila ran D IV IS IO N M E M O R A N D U M N o.3 T 3 ^ s. 2 0 1 6 R e p u b lic o f th e P h ilip p in e s D e p a rtm e n t o f E d u catio n R e g io n V II, C e n tra l V isa y a s D IV IS IO N O F B O H O L

More information

History 152 World History II

History 152 World History II 1 History 152 World History II 11:00-12:15 TR Prof. Michael Bitter UCB 245 Office: UCB 352 Fall 2012 Tel.: 974-7466 Section 001 Email: bitter@hawaii.edu Office Hours: Mon. & Wed.: 2:00-2:50 Tues. & Thurs.

More information

THE EFFECT Of SUSPENSION CASTING ON THE HOT WORKABILITY AND MECHANICAL PROPERTIES OF A IS I TYPE STAINLESS STEEL

THE EFFECT Of SUSPENSION CASTING ON THE HOT WORKABILITY AND MECHANICAL PROPERTIES OF A IS I TYPE STAINLESS STEEL THE EFFECT Of SUSPENSION CASTING ON THE HOT WORKABILITY AND MECHANICAL PROPERTIES OF A IS I TYPE 3 1 0 STAINLESS STEEL A LISTAIR GEORGE SANGSTER FORBES A D i s s e r t a t i o n s u b m i tte d t o th

More information

NATO and Canada, : The Tight-Lipped Ally

NATO and Canada, : The Tight-Lipped Ally Canadian Military History Volume 24 Issue 2 Article 9 11-23-2015 NATO and Canada, 1990-1993: The Tight-Lipped Ally Ian Weatherall Recommended Citation Ian Weatherall (2015) "NATO and Canada, 1990-1993:

More information

BIRLA ERICSSON OPTICAL LIMITED

BIRLA ERICSSON OPTICAL LIMITED OPTICAL LIMITED ANNUAL REPORT 2012-13 BOARD OF DIRECTORS MR.HARSH V. LODHA MR.D.R.BANSAL MR.MAGNUS KREUGER [ALTERNATE MR.DINESH CHANDA] MR.MATS O.HANSSON [ALTERNATE MR.S.K.DAGA] MR.R.C.TAPURIAH DR.ARAVIND

More information

History 151 World History I

History 151 World History I 1 History 151 World History I 11:00-12:15 TR Prof. Michael Bitter UCB 118 Office: UCB 352 Fall 2011 Tel.: 974-7466 Section 005 Email: bitter@hawaii.edu Office Hours: 2:00-3:00 MW 10:00-11:00 TR 2:00-3:00

More information

@ *?? ^ % ^ J*

@ *?? ^ % ^ J* M A R IN E & O F F S H O R E C A B L E S m m @ B O g g B @ *?? @-@ ^ % ^ - @* J* M a r in e a n d o ffs h o re s ta n d a rd s a n d te s ts IE C 6 0 0 9 2-3 50 \ le ctrica l in sta llatio n s in s h ip

More information

Operation Manual for Automatic Leveling Systems

Operation Manual for Automatic Leveling Systems Operation Manual for Automatic Leveling Systems 11/12 Power Gear #82-L0379 Rev. 0E Operation Manual for Automatic Leveling Systems with Touch Pad # 140-1226 and Control Box # 140-1229 Contents Before You

More information

M. H. DALAL & ASSOCIATES C H ARTERED ACCOUNTANTS

M. H. DALAL & ASSOCIATES C H ARTERED ACCOUNTANTS M. H. DALAL & ASSOCIATES C H ARTERED ACCOUNTANTS 301/308, Balaji D arshan, Tilak R oad, Santacruz (W ), M um bai - 400 054. Phone : 26494807 : 26490862 E-m ail: m hdalal@ gm ail.com W ebsite: w w w.dalalgroup.in

More information

1980 Annual Report / FEDERAL R ESER V E BA N K OF RICHMOND. Digitized for FRASER Federal Reserve Bank of St.

1980 Annual Report / FEDERAL R ESER V E BA N K OF RICHMOND. Digitized for FRASER   Federal Reserve Bank of St. 1980 Annual Report / FEDERAL R ESER V E BA N K OF RICHMOND IS S N 0164-0798 L IB R A R Y OK C O N G R E SS C A T A L O G C A R D N U M B E R : 16-72o4 Additional <

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

Sodium-Initiated Polymerization of Alpha- Methylstyrene in the Vicinity of Its Reported Ceiling Temperature

Sodium-Initiated Polymerization of Alpha- Methylstyrene in the Vicinity of Its Reported Ceiling Temperature Western Michigan University ScholarWorks at WMU Dissertations Graduate College 8-1976 Sodium-Initiated Polymerization of Alpha- Methylstyrene in the Vicinity of Its Reported Ceiling Temperature Shuenn-long

More information

Report Documentation Page

Report Documentation Page % &('()*! "$# +-,/. 0214365*798;:@:(021BAC3ED=1FG1H3@D=1H3?IJ86KL3=1M!KON$:IPKOQ?3SR3@0?KO3@1 TVUXWY Z[VY \

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

ST 602 ST 606 ST 7100

ST 602 ST 606 ST 7100 R e s to n Preserve Ct P r e s e rve Seneca Rd S h e r m a n ST 193 B2 Georgetown Pike B6 B8 Aiden Run Ct B9 Autumn Mist Ln B12 B11 B16 B15 B13 B17 Shain Ct B19 B21 B26 B24 B23 N o r th fa lls B28 B27

More information

A Comparison of Two Methods of Teaching Computer Programming to Secondary Mathematics Students.

A Comparison of Two Methods of Teaching Computer Programming to Secondary Mathematics Students. Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1983 A Comparison of Two Methods of Teaching Computer Programming to Secondary Mathematics Students.

More information

UNITED STATES SECURITIES AND EXCHANGE COMMISSION Washington, DC FORM 8-K. Current Report

UNITED STATES SECURITIES AND EXCHANGE COMMISSION Washington, DC FORM 8-K. Current Report UNITED STATES SECURITIES AND EXCHANGE COMMISSION Washington, DC 20549 FORM 8-K Current Report Pursuant to Section 13 or 15(d) of the Securities Exchange Act of 1934 Date of Report (Date of earliest event

More information

7.2 P rodu c t L oad/u nload Sy stem s

7.2 P rodu c t L oad/u nload Sy stem s 7.2 P rodu c t L oad/u nload Sy stem s The 10" or 12" augers, or the 10" conveyor are high capacity load/unload systems with hydraulic controls to manoeuvre and operate. The hydraulic assist allows the

More information

Texas Student Assessment Program. Student Data File Format for Student Registration and Precoding

Texas Student Assessment Program. Student Data File Format for Student Registration and Precoding Texas Student Assessment Program 2018 Student Data File Format for Student Registration and Precoding 2 2018 Student Data File Format for Student Registration and Precoding Submission Schedule of Student

More information

Rule-Governed Behavior in Preschool Children

Rule-Governed Behavior in Preschool Children Western Michigan University ScholarWorks at WMU Master's Theses Graduate College 12-1985 Rule-Governed Behavior in Preschool Children Cassandra Ann Braam Western Michigan University Follow this and additional

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

Compulsory Continuing Education for Certified Public Accountants: a Model Program for the State of Louisiana.

Compulsory Continuing Education for Certified Public Accountants: a Model Program for the State of Louisiana. Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1975 Compulsory Continuing Education for Certified Public Accountants: a Model Program for the State

More information

MySQL 5.1. Past, Present and Future. Jan Kneschke MySQL AB

MySQL 5.1. Past, Present and Future. Jan Kneschke MySQL AB MySQL 5.1 Past, Present and Future Jan Kneschke MySQL AB Agenda Past S Q L T re e s m e e ts D y n a m ic S Q L P re s e n t E v e n ts P a rtitio n in g F u tu re V e rtic a l P a rtitio n in g About

More information

Taiwan Radio Occultation Process System (TROPS)

Taiwan Radio Occultation Process System (TROPS) Taiwan Radio Occultation Process System (TROPS) Cheng-Yung Huang, Wen-Hao Yeh, Kun-Lin Chen National Space Organization, National Applied Research Laboratories Shu-Ya Chen, Jing-Mei Wu, Hsiu-Wen Li, Kuo-Hsin

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

Applied Tape Techniques for Use With Electronic Music Synthesizers.

Applied Tape Techniques for Use With Electronic Music Synthesizers. Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1974 Applied Tape Techniques for Use With Electronic Music Synthesizers. Robert Bruce Greenleaf Louisiana

More information

Matador Resources Company (Exact name of registrant as specified in its charter)

Matador Resources Company (Exact name of registrant as specified in its charter) UNITED STATES SECURITIES AND EXCHANGE COMMISSION Washington, D.C. 20549 FORM 8-K CURRENT REPORT Pursuant to Section 13 or 15(d) of the Securities Exchange Act of 1934 Date of Report (Date of Earliest Event

More information

Memorial to William Taylor Thom, Jr.

Memorial to William Taylor Thom, Jr. Memorial to William Taylor Thom, Jr. 1891-1979 S H E L D O N J U D S O N Department o f Geological and Geophysical Sciences, Princeton University, Princeton, New Jersey 08544 W illia m T a y lo r T h o

More information

INCOME TAXES IN ALONG-TERMMACROECONOMETRIC FORECASTING MODEL. Stephen H. Pollock

INCOME TAXES IN ALONG-TERMMACROECONOMETRIC FORECASTING MODEL. Stephen H. Pollock INCOME TAXES IN ALONG-TERMMACROECONOMETRIC FORECASTING MODEL. by Stephen H. Pollock Dissertation submitted to the Faculty of the Graduate School of the University of Maryland in partial fulfillment of

More information

IV IS IO N O F B O H OL \ % 1J \

IV IS IO N O F B O H OL \ % 1J \ R ep u b li f th e P h ilip p in e s D E P A R T M E N T O F E D U C A T ION \ eg i n V II, C e n trl V isy s @ % gjs i* W H T \ D tje D C ity f T g b ilrn d e p rtm ent 1 f ed u ti n IV IS IO N O F B

More information

INFORMATION TO USERS

INFORMATION TO USERS INFORMATION TO USERS This material was produced from a microfilm copy of the original docum ent. While the m ost advanced technological means to photograph and reproduce this docum ent have been used,

More information

PRODUCT DATA SHEET MODEL: SUK-0084

PRODUCT DATA SHEET MODEL: SUK-0084 PRODUCT DATA SHEET MODEL: SUK-0084 SPECIFICATIONS: MODEL NUMBER: SUK-0084 PUMP DESIGN: Positive displacement 3 chamber diaphragm pump CHECK VALVE: No CAM: 2 MOTOR: 4 Pole Induction VOLTAGE: 230VAC PRESSURE

More information

Vlaamse Overheid Departement Mobiliteit en Openbare Werken

Vlaamse Overheid Departement Mobiliteit en Openbare Werken Vlaamse Overheid Departement Mobiliteit en Openbare Werken Waterbouwkundig Laboratorium Langdurige metingen Deurganckdok: Opvolging en analyse aanslibbing Bestek 16EB/05/04 Colofon Ph o to c o ve r s h

More information

UNITED STATES SECURITIES AND EXCHANGE COMMISSION WASHINGTON, DC FORM 8-K CURRENT REPORT

UNITED STATES SECURITIES AND EXCHANGE COMMISSION WASHINGTON, DC FORM 8-K CURRENT REPORT UNITED STATES SECURITIES AND EXCHANGE COMMISSION WASHINGTON, DC 20549 FORM 8-K CURRENT REPORT PURSUANT TO SECTION 13 OR 15(D) OF THE SECURITIES EXCHANGE ACT OF 1934 Date of report (Date of earliest event

More information

NUMERICAL SIMULATION OF MHD-PROBLEMS ON THE BASIS OF VARIATIONAL APPROACH

NUMERICAL SIMULATION OF MHD-PROBLEMS ON THE BASIS OF VARIATIONAL APPROACH NUMERICAL SIMULATION OF MHD-PROBLEMS ON THE BASIS OF VARIATIONAL APPROACH V.M. G o lo v izn in, A.A. Sam arskii, A.P. Favor s k i i, T.K. K orshia In s t it u t e o f A p p lie d M athem atics,academy

More information

SCHOOLS DIVISION OFFICE OF KABANKALAN CITY

SCHOOLS DIVISION OFFICE OF KABANKALAN CITY D e fje D DEPARTMENT \ g OF EDUCATION Republic of the Philippines Department o f Education Negros Island Region SCHOOLS DIVISION OFFICE OF KABANKALAN CITY City o f Kabankalan Tel.No (034) 471-2003 Fax

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

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

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

MAHARASHTRA STATE BOARD OF TECHNICAL EDUCATION

MAHARASHTRA STATE BOARD OF TECHNICAL EDUCATION (ISO/IEC - 7-5 Certiied) Page No: /6 WINTER 5 EXAMINATION MODEL ANSWER Subject: ENGINEERING MATHEMATICS (EMS) Subject Code: 76 Important Instructions to eaminers: The model answer shall be the complete

More information

Photo. EPRI s Power System and Railroad Electromagnetic Compatibility Handbook

Photo. EPRI s Power System and Railroad Electromagnetic Compatibility Handbook Photo EPRI s Power System and Railroad Electromagnetic Compatibility Handbook Brian Cramer Project Manager Transmission and Substations bcramer@epri.com 815/478-5344 Problem Periodic false activation of

More information

Olivet College Fifteenth Annual Catalog

Olivet College Fifteenth Annual Catalog Olivet Nazarene University Digital Commons @ Olivet Catalog Academic Affairs Office 1923 Olivet College Fifteenth Annual Catalog 1923-1924 Olivet Nazarene University Olivet Nazarene University Follow this

More information

Product Type: DC Stock: Stock Description: 1/3HP.1750RPM.MSS56C.TEFC.90V.CONT.40C.1.0SF.RIGID C.DC NEMA.C42D17FK4C.

Product Type: DC Stock: Stock Description: 1/3HP.1750RPM.MSS56C.TEFC.90V.CONT.40C.1.0SF.RIGID C.DC NEMA.C42D17FK4C. Catalog # : 098004.00 Model: C42D17FK4 Product Type: DC Stock: Stock Description: 1/3HP.1750RPM.MSS56C.TEFC.90V.CONT.40C.1.0SF.RIGID C.DC NEMA.C42D17FK4C Quote: Price: Date: 05/18/2016 Catalog # : 098004.00

More information

SINTERING AND CHARACTERISATION OF NANO SIZED YTTRIA-STABILISED ZIRCONIA. P r e p a r e d b y. Muhammad Hasanuzzaman, B.Sc. (Eng)

SINTERING AND CHARACTERISATION OF NANO SIZED YTTRIA-STABILISED ZIRCONIA. P r e p a r e d b y. Muhammad Hasanuzzaman, B.Sc. (Eng) SINTERING AND CHARACTERISATION OF NANO SIZED YTTRIA-STABILISED ZIRCONIA P r e p a r e d b y Muhammad Hasanuzzaman, B.Sc. (Eng) A T h e s i s S u b m i t t e d f o r t h e f u l f i l l m e n t o f t h

More information

Feasibility Analysis, Dynamics, and Control of Distillation Columns With Vapor Recompression.

Feasibility Analysis, Dynamics, and Control of Distillation Columns With Vapor Recompression. Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1981 Feasibility Analysis, Dynamics, and Control of Distillation Columns With Vapor Recompression.

More information

A BEGRIFFSSCHRIFT FOR SENTENTIAL LOGIC. John EVENDEN 1. INTRODUCTION

A BEGRIFFSSCHRIFT FOR SENTENTIAL LOGIC. John EVENDEN 1. INTRODUCTION A BEGRIFFSSCHRIFT FOR SENTENTIAL LOGIC John EVENDEN 1. INTRODUCTION The title of this essay is probably a misnomer, but the aim is to achieve f o r sentential lo g ic something v e ry simila r t o Frege's

More information