Some Equivalent Forms of Bernoulli s Inequality: A Survey *
|
|
|
- Karen Small
- 5 years ago
- Views:
Transcription
1 Ale Mthets htt://oog/436/34746 Pulshe Ole Jul 3 (htt://wwwsog/joul/) Soe Euvlet Fos of Beoull s Ieult: A Suve * u-chu L Cheh-Chh eh 3 Detet of Ale Mthets Ntol Chug-Hsg Uvest Tw Detet of Mthets Ntol Cetl Uvest Tw 3 Detet of Ifoto Mgeet Lughw Uvest of See Teholog Tw El: l@ethhueutw ghe@thueutw gho85@glo eeve M 3; evse Jue 3; ete Jue 9 3 Coght 3 u-chu L Cheh-Chh eh Ths s oe ess tle stute ue the Cetve Coos Attuto Lese whh ets uestte use stuto eouto eu ove the ogl wo s oel te ABSTACT The uose of ths e s to l soe ow eultes whh e euvlet to Beoull s eult Kewos: Beoull s Ieult; oug s Ieult; Jese s Ieult; Höle s Ieult; Cuh s Ieult; Mows s Ieult; Shlöh s Ieult; AGM Ieult; Josthl s Ieult; Euvlet Itouto Bse o the theo of eultes lssl eultes ot ol oote the eveloet of the eult theo ut lso le to ltos ue thets le thets Beoull s eult s oe of the ost stgushe eultes I ths e ew oof of Beoull s eult v the ese oet s gve Soe stegthee fos of Beoull s eult e estlshe Moeove soe euvlet eltos etwee ths eult othe ow eultes e tettvel le The ogzto of ths e s s follows: I Seto ew oof of Beoull s eult es of the oet of est s se I Seto 3 soe stegthee fos of Beoull s eult e estlse I Seto 4 we l soe ow eultes whh e euvlet to Beoull s eult I Seto 5 we ollet soe vts of oug s eult whh e euvlet to Beoull s eult Fo elte esults we efe to [-35] Peles I oe to olete these tss we ee the efto soe s esults of the ove futo s follows: Defto Let f : I e futo whee I s tevl of * Dete to the esete Pofesso Huo Mu ) Suose tht P Q e two ots o the gh of f f the ho PQ ot elow the PQ of the gh of f the we s tht f s ove futo o I Tht s fo two ot I t f t t tf t f () the f s ove futo o I We s tht f s lle ove o I f f s ove o I If fo two ots I wth t f f t t tf t f the we s tht s sttl ove futo o I ) I s s to e ot ove o J-ove o I f fo two ots I f f f () It s well-ow ft tht eve ove futo o tevl s otuous; f f s -ot ove otuous o tevl I the t s ove o I The followg Jese s eult e show the thetl uto etl Le (Jese s eult [3] ge 3) Let f e ove futo o I The fo wth fo I Coght 3 Ses
2 -C LI C-C EH 7 f f (3) If f s sttl ove the (3) s sttl uless the e ll etll Le 3 Let f : I e futo The the followg stteets e euvlet: ) f s sttl ove o I ) Fo two stt ots I I stsfg f f f ; 3) Fo two stt ots I I stsfg f f f Poof ) ) Let I e stt let e t If z I the z s etwee z It follows fo the stt ovet of f o I tht f f z f Hee ) hols ) 3) Let I e stt let e t If z I the z oth z e stt B the ssuto of () we hve f f f z It follows fo tht 3) hols 3) ) Let I e stt let e t If z the z It follows fo the ssuto of 3) tht f f f z Ths ove ) hols Thus the oof s olete Net we wll ove Beoull s eult es of the oet of est wthout ffeetto o tegto Le 4 foll wth (4) The eult s ovous fo se = o fo se o Poof Let E : Cl : E s ese It suffes to show tht E stsfes the followg thee oetes ) E ) If E the E 3) If E the E E Let e t wth The Thus So 4 < Ths oves ) hee E s oet If E the Ths oves ) Net f e suh tht E the fo eve wth Ths oves the fst t of 3) O the othe h t follows fo E tht tht Theefoe (5) Thus we olete the oof of 3) Se )-3) l E fo Coght 3 Ses
3 7 -C LI C-C EH 3 Theefoe E ust e ese Fll f s t E wth the fo eve s Ths oves Sll f fo s t E wth the fo eve s Ths oves fo Theefoe fo eve we hve fo wth It follows fo (5) g tht (4) hols Ths oletes the oof Cooll 5 The followg stteets e euvlet: ) l s sttl ove o ; ) fo ll > wth fo ll ; 3) oug s eult hols tht s X X whee X > wth X wth ; 4) (4) hols Poof hols tht s foll wth foll fo ll wth fo ll fo ll wth fo ll hols tht s fo ll wth fo ll l s sttl ove o tht s hols X X fo X wth X fo ll wth ; hols The eult of oug s eult s le fo se X wth X Ths oletes the oof Net we ove soe euvlet esults whh e elte to l : Le 6 Fo the followg stteets e euvlet: ) l s sttl ove o ; ) l s sttl ove o ; 3) l s sttl ove o Poof Clel ) ) 3) Now we ove 3) ) ) ) Let t e wth e t Se let l t t l t l we hve l tt l tl t l l l Thus f t s sll suh tht t t we ot tht 3) les ) Sll f t s eough lge so tht t t we ot tht ) les ) Ths oletes the oof Le 7 Let stsfg = e ll ostve o ll egtve If fo ll wth the Poof Ths le s tue fo ssuto Suose tht ths le hols fo Let If the lel the oluso hols Now we ssue Se e ll ostve o ll egtve we see tht Theefoe Coght 3 Ses
4 -C LI C-C EH 73 Ths oletes ou oof 3 Vts of Beoull s Ieult I ths seto we estlsh soe vts of Beoull s Ieult Se l s sttl ove sttl esg o ts vese futo e s sttl ove sttl esg Usg Le 7 we hve the followg Theoe 3 The followg eultes e euvlet: l s sttl ove o ; whee tht s whee < stsf whee whee ; tht s whee stsf whee tht s whee ; whee 3 ; whee ; ; whee ; tht s whee ; whee 3 3 tht s whee ; whee whee ; whee ; tht s whee ; whee > ; whee ; whee ; whee ; whee ; whee ; whee ; Coght 3 Ses
5 74 -C LI C-C EH whee > ; whee ; whee ; whee < ; whee ; 9 whee Poof Let : whee the ove futo : hols : l l l foll wthfoll fo ll fo ll fo ll fo ll z z fo ll z fo ll hols Let WLOG we ssue > > The : s sttl Now we ssue hols fo Set We hve fo wth Ths oves Let It follows fo ove guet the uto ssuto tht s ovous Moeove t follows fo Le 7 tht : B hols Coght 3 Ses
6 -C LI C-C EH 75 hols : hols whee whee whee : whee hols : hols whee whee : whee hols : hols 3 3 whee whee > > hols 3 : It follows fo hols whee 3 tht whee hols : hols whee whee > whee whee Coght 3 Ses
7 76 -C LI C-C EH 4 hols : hols 4 4 hols : hols whee whee whee > whee whee whee : hols 3 It follows fo 5 : whee tht hols whee whee hols : hols whee whee whee hols 5 : hols whee whee whee : whee : 5 Coght 3 Ses
8 -C LI C-C EH 77 3 hols : hols 7 7 whee < whee whee : hols : hols 7 7 : > whee < whee whee : hols : hols 8 8 : < < whee whee whee : hols : hols 8 8 : whee < whee whee : hols : hols : > whee whee whee : : Coght 3 Ses
9 78 -C LI C-C EH 3 hols : hols whee whee : whee 3 hols Ths ove ou Theoe B Theoe 3 we hve the followg Cooll 3 Let e ostt If the the followg thee eultes e euvlet: ) ) 3) Poof Clel t follows fo Theoe 3 tht hol; ) hols ) hols 3) hols 3 4 M esults hol; hol Now we stte ove soe eultes whh e euvlet to eh othe the followg t 3 Theoe 4 Let whee s ostve tege The the followg soe stteets e euvlet: whee ; whee ; whee ; whee ; whee whee ; ; whee o ; whee ; whee ; whee ; 3 3 whee ; whee o ; 3 whee ; whee ; 3 Coght 3 Ses
10 -C LI C-C EH 79 3 whee ; whee ; whee whee 7 l s ove o 7 whee ; 8 9 ; whee ; whee ; Thus 3 whee ; ; ; whee hee s esg futo of whee ; whee < < ; hee whee o ; whee ; 4 whee ; 4 f 4 (Cuh s eult); whee ; M M whee s 5 6 s M wth (Höle s eult); : whee whee > > < ; Coght 3 Ses
11 8 -C LI C-C EH Hee I tul s 6 s = = s (Shlöh s eult); 6 fo ; = 6 fo ; 7 t M M I geel ; 7 7 (Mows eult) (Mows eult) hee t t t t t ; (AGM eult) Sho s eult: 7 (see [7]) G j A j j j j l l ; j j whee j A ; j see the followg fgue: whee j j G 7 j j j j e 7 G G G AA A j j j j A G A : G whh s euvlet to G H ; whee 8 8 ; 8 whee e ostve teges s tol ue; 8 whee ostve tege ; 8 whee ; whee e 8 9 e e e ; s (sttl) esg o ; j ; tht s Coght 3 Ses
12 -C LI C-C EH s (sttl) esg o ; e s (sttl) eesg o ; whee e l ; e whee t hs followg soe vts: e whee l whee 3 e whee 4 e o e whee 5 e 6 whee e whee l 7 l 8 9 whee whee l whee l whee o e whee l whee 3 l whee e e e whee 4 5 l e whee ; l whee tht s l whee ; l whee ; l whee o ; tht s l ; e e whee ; tht s e e whee ; whee ; whee e e ; tht s (Josthl s eult); Coght 3 Ses
13 8 -C LI C-C EH ( ) > whee o Poof Tg Cooll 3 we see tht e euvlet Sll elg Cooll we get tht 3 e euvlet Hee t follows fo Theoe 3 tht e euvlet If the lel wth = 3 follows tg = + Hee : We see tht hols hols ff > Sll we ove 3 3 follows fo Theoe follows fo Theoe 3 too : Let stsf 4 B we see tht Thus t follows fo tht Hee 4 hols elg esetvel Sll we ove 3 : Let : whee the tht j It follows fo j j Hee j j j Ths oletes the oof of 3 6 see H et ([8] Theoe 9 6) A tht : It follows fo 8 A A A A A A A A Thus A A (see Mlgs [8] o oo [8]) Hee A A A G 8 hols 8 : Tg Theefoe we see tht Hee Dvg oth ses we get 8 s le : We show the thetl uto o If s ovous Suose hols fo wth Set = + If t s es to see tht eh the ssuto hee Theefoe hols Assue Se we hve j Coght 3 Ses
14 -C LI C-C EH 83 Thus hols : Tg we see tht 8 hols 3 3 : Clel 3 3 Let suh tht : If : the t follows fo tht (5) B Clel B (5) 3 Thus s ove : Let If the : B 3 Ths oletes the oof of : Wthout loss of geelt we ssue tht > t Se l s sttl esg t t tt tl t l l t t B the efto of the ove futo : Tg see tht hols : Let The : Let The 7 9 : Let The Hee so 3 4 elg 7 7 : If the 3 Thus s ove 4 Sll we ove : Let s 3 6 elg 4 3 the s s s 3 7 we Coght 3 Ses
15 84 -C LI C-C EH Hee s s M M fo s s M M fo s s f M Ms s If the M Ms s Hee the M M fo s s tht It follows tg 6 hols Sll we ove : Let A elg A 4 esetvel fo we ot 4 thus we olete the oof 4 4 follows tg = = : Let f fo 4 tht fo F The t follows F = = F Thus l F s ove o hee l F s ove o Hee fo l F l F l F whh les F F F Lettg the oth ses of the ove eult F F Ths shows tht 4 hols see L Shw [5] 4 5 : Let > : : whee B 4 tht It follows fo (6) jj j Hee 5 hols : Let 5 4 The = = j Thus 5 whee e efe s ove Hee Thus = = j j j= j= j j j j j j Ths oletes the oof of 4 j j (6) 6 4 : Tg s elg follows tg : Tg s 6 fo elg 6 fo we ot 4 6 fo thus Coght 3 Ses
16 -C LI C-C EH 85 we olete the oof 4 6 : see 55 of Mtov [9] Sll we ove follows tg s follows tg s follows tg s follows tg follows tg (wth 7 7 : elg 7 3 tht oth ) (wth we ove 7 ) e le Sll we ove 7 7 : Tg t t t we see esl 7 7 Hee 7 3 e euvlet : Wthout loss of geelt we ssue tht A j 3 Ths oletes the oof of 7 A A A A A A A A A A A A A A A A A A j j A A A j Thus 7 7 : Tg the tul logth the oth ses of 7 we get 7 7 we get : Tg 7 we get G H 7 7 : Tg 7 we see tht 7 hols tul logth of the oth ses of e : Tg : Tg 7 7 e we get e 7 e 7 usg the followg fgue we get 7 Covesel eletg the 8 3 Coght 3 Ses
17 86 -C LI C-C EH s le : Tg 8 we get 8 8 follows tg (see [33]): It follows fo tht : Let The 8 Thus Clel f the the ove eult hols too 8 : Clel f the hols Suose tht 8 hols fo Thus fo f the Hee Ths 8 e 8 7 Thus : If the 7 Thus hols 9 : 9 e olete the oof of e Theefoe e 8 el Let The Thus hols tht s whee hols Theoe 9 : 9 hols tht s e l e s sttl esg futo o whee hols Theoe : 9 hols tht s : s sttl esg futo o s sttl esg futo o 8 Coght 3 Ses
18 -C LI C-C EH 87 9 hols Thus 9 s sttl esg futo o s sttl esg futo o 9 9 e euvlet 9 9 : Fo ll se 9 9 I tul fo ll > e euvlet 4 4 ohes to s Thus e l 9 9 we get 9 see [] 9 : B the fst eult of 9 Hee e e B the seo eult of 9 Hee e e It follows fo eh fo tht fo e If o the lel hols Ths oletes the oof of 9 Hee : B e e tht s the fst eul- t of 9 hols Net Hee f the e e whee : Thus the seo eult of 9 hols s le 8 : Tg of A l A A whee Sug ths eultes we get 8 hols 8 : e e A A G A A see Bulle ([3] 7) o Kug ([4] 33) : B follows tg follows tg l l e Thus e e elg we oletes the oof see Clou Dh ([6] 3) e s le 8 : B G e e G Coght 3 Ses
19 88 -C LI C-C EH A G G Thus 8 Hee e e e hols see [3] [9] 8 9 e euvlet we lso efe to [] : Wthout loss of geelt we 8 ssue tht Hee B Thus 8 hols e l l 8 : Fo > t follows fo e e e Tg e e 8 see H et ([8] 4-4) o Wg Su Wg [33] : Tg Hee tht 8 : Tg A we get A A Hee A A A G Ths oletes the oof of 8 see Bulle ([3] 98) o Kug ([4] 33) follows tg = fo = 8 7 : B 8 l l l Hee l s ot ove o Se l s otuous o l s ove futo o Thus 7 hols We lso ove 8 8 usg the thetl uto Thus ou oof s olete 5 Othe Euvlet Fos of Beoull s Ieult I ths seto we shll ollet soe vts of oug eult whh s euvlet to the Beoull s eult Theoe 5 Let s e ostve ues fo If whee the el ues stsf the the followg soe eultes e euvlet: whee ; whee ; whee ; whee ; whee ; whee ; t wheet ; t t wheet ; t t wheet ; t Coght 3 Ses
20 -C LI C-C EH 89 t 3 t whee t ; t 3 t whee t ; 3 t whee ; t t t t whee t ; t t whee t ; t t t 5 5 whee ; t wheet ; t t wheet ; t 5 t wheet ; t 6 s whee ; s 6 s whee ; s 6 s whee ; s t t whee ; t t whee ; t t whee ; whee ; l e whee ; e whee ; whee ; whee ; whee ; Coght 3 Ses
21 9 -C LI C-C EH h h h h whee h ; h h h h h h whee ; h h h h h h h whee ; h h whee ; whee thee ests etl oe of s ostve the othe e egtve; 3 whee stsfg ; 3 whee stsfg ; Poof Clel euvlet s le e vt of : elg 8 Hee Thus : Let Ths oletes ou oof esetvel Hee t t esetvel we get whee The whee : Fo ll stsfg Ths ove the oof of follows tg : Let see lso Su [3] stsf The fo e Coght 3 Ses
22 -C LI C-C EH 9 Thus hols follows elg s le Sll we ove If the h h h h e ove sll follows usg the thetl uto Sll t : elg t whee follows elg : elg t t whee : elg s : elg s : elg whee h : elg h h esetvel 4 esetvel we get t t esetvel Covesel esetvel whee Sll we ove esetvel we get 6 esetvel we get 6 t t esetvel we get 7 Sll we ove 7 h h : It follows fo Theoe 3 tht 3 elg 3 tht we ssue tht It follows fo esetvel we get whee e euvlet Wthout loss of geelt 3 esetvel whee 3 hols whee 5 3 : elg t : elg t : elg t t we get 3 whee t 3 esetvel we get 5 t we get 3 follows elg t 6 follows tg whee t 5 s 6 whee Sll we ove 3 esetvel whee Coght 3 Ses
23 9 -C LI C-C EH get 6 6 follows elg s s esetvel 8 9 : Se l l l e 9 9 : Tg 9 we get : Lettg u s s 9 e l Ths oletes the oof of 9 9 s vt of follows tg h e s e se s theefoe 8 we get 9 A the tg sl we see Su [3] whee : B Theoe 3 3 e euvlet It suffes to show 3 elg We ssue the elg 3 tg hols 3 : It follows fo esetvel t follows fo tht 3 tht elg we olete ou oof Sll we ove e le e 5 Fo eult 6 we efe to Isuo Tog [3] Fo eult 8 we efe to [3] Fo eult 9 9 we efe to Su [3] we efe to Kug Fo eult [4] Fo eult Fo eult 3 we efe to [34] we efe to Su [3] e 53 Thee e vts of Höle s eult Shlöh s eult AGM eult Mows s eult so o we ot the etl EFEENCES [] E F Beeh Bell Ieult 4th Eto Sge-Velg Bel 984 [] E F Beeh W Wle Geel Ieultes III Bhäuse Velg Bsel 983 [3] P S Bulle Hoo of Mes The Ieultes Kluwe Ae Pulshes Doeht 3 o:7/ [4] P S Bulle D S Mtovć P M Vsć Mes The Ieultes D eel Pulshg Co Doeht 95 [5] P S Bulle A Chte o Ieultes Southest As Bullet of Mthets Vol [6] M J Clou B C Dh Ieultes wth Altos to Egeeg Sge Velg New o 998 [7] C Geogs O the Ieult fo the Athet Geoet Mes Mthetl Ieultes Altos Vol [8] G H J E Lttlewoo G Pól Ieultes Eto Cge Uvest Pess Cge 95 [9] Z Ho Note o the Ieult of the Athet Geoet Mes Pf Joul of Mthets Vol 43 No [] J How J How Euvlet Ieultes The College Mthets Joul Vol 9 No [] K Hu Soe Poles of Alt Ieultes ( Chese) Wuh Uvest Pess Wuh 3 [] C A Iftozz A Itouto to eltos og Ieultes Notes of the Ae Mthetl Soet Vol 4 97 A98-A8 [3] S Isuo M Tog Estto Höle s Te Ieult Mthetl Ieultes Altos Vol [4] J Kug Ale Ieultes ( Chese) 3 Eto Shog See Teholog Pess Shog Coght 3 Ses
24 -C LI C-C EH 93 4 [5] -C L S Shw A Poof of Höle s Ieult Usg the Cuh-Shwz Ieult Joul of Ieultes Pue Ale Mthets Vol 7 No 6 [6] C K L Cove Futos Jese s Ieult Legee Tsfoto ( Chese) Mthe Ae S Vol [7] C K L The Essee Sgfe of Cuh- Shwz s Ieult ( Chese) Mthe Ae S Vol [8] L Mlg Wh Höle s Ieult Shoul Be Clle oges Ieult Mthetl Ieultes Altos Vol [9] A W Mshll I Ol Ieultes: Theo of Mjozto Its Altos Ae Pess New o 979 [] D S Mtovć Alt Ieultes Sge-Velg Bel 97 [] D S Mtovć J E Pečć A M F Clssl New Ieultes Alss Klue Ae Pulshe Doeht 993 o:7/ [] D S Mtovć J E Pečć Beoull s Ieult eot el Colo Mteto Pleo Vol 4 No [3] D J New Athet Geoet Ieult The Ae Mthetl Mothl Vol 67 No o:37/3946 [4] N O Oze M K Ao Ieultes ( Jese) 3 Eto M Shote Too 967 [5] J E Pečć O Beoull s Ieult A Nu Ujet Bos Heegov Oelj Po Mt Nu Vol [6] J Pečć K B Stols Cle s Ieult: Hsto New Geelztos Aeutoes Mthete Vol 6 No o:7/s56 [7] J Pečć S Vše A New Poof of the Athet Me The Geoet Me Ieult Joul of Mthetl Alss Altos Vol 5 No o:6/j [8] J oo Soe New Poofs fo the AGM Ieult Mthetl Ieultes Altos Vol 7 No [9] N Shuege A Coote Aoh to the - GM Ieult Mthets Mgze Vol 64 No o:37/69837 [3] S-C Sh Covet Dl Uvest of Teholg Pess Dl [3] X H Su O the Geelze Höle Ieultes Soohow Joul of Mthets Vol [3] C L Wg Ieultes of the o-poovu Te fo Futos The Altos Joul of Mthetl Alss Altos Vol No o:6/-47x(84)99- [33] X T Wg H M Su F H Wg Ieultes Theo Methos ( Chese) He Euto Pulto Zhegzhou 967 [34] J We W Wg H Zhou Z g A Clss of Cl Ieultes of Jous Te ( Chese) Joul of Chegu Uvest Vol [35] C X Xue Isolto Eteso of Beoull Ieultes ( Chese) Joul of Gsu Euto College Vol 3 No Coght 3 Ses
A Dynamical Quasi-Boolean System
ULETNUL Uestăţ Petol Gze Ploeşt Vol LX No / - 9 Se Mtetă - otă - Fză l Qs-oole Sste Gel Mose Petole-Gs Uest o Ploest ots etet est 39 Ploest 68 o el: ose@-loesto stt Ths e oes the esto o ol theoetl oet:
Difference Sets of Null Density Subsets of
dvces Pue Mthetcs 95-99 http://ddoog/436/p37 Pulshed Ole M (http://wwwscrpog/oul/p) Dffeece Sets of Null Dest Susets of Dwoud hd Dsted M Hosse Deptet of Mthetcs Uvest of Gul Rsht I El: hd@gulc h@googlelco
Moments of Generalized Order Statistics from a General Class of Distributions
ISSN 684-843 Jol of Sttt Vole 5 28. 36-43 Moet of Geelzed Ode Sttt fo Geel l of Dtto Att Mhd Fz d Hee Ath Ode ttt eod le d eel othe odel of odeed do le e ewed el e of geelzed ode ttt go K 995. I th e exlt
The formulae in this booklet have been arranged according to the unit in which they are first
Fomule Booklet Fomule Booklet The fomule ths ooklet hve ee ge og to the ut whh the e fst toue. Thus te sttg ut m e eque to use the fomule tht wee toue peeg ut e.g. tes sttg C mght e epete to use fomule
A convex hull characterization
Pue d ppled Mthets Joul 4; (: 4-48 Pulshed ole My 4 (http://www.seepulshggoup.o//p do:.648/.p.4. ove hull htezto Fo Fesh Gov Qut Deptet DISG Uvesty of Se Itly El ddess: fesh@us.t (F. Fesh qut@us.t (G.
AN ALGEBRAIC APPROACH TO M-BAND WAVELETS CONSTRUCTION
AN ALGEBRAIC APPROACH TO -BAN WAELETS CONSTRUCTION Toy L Qy S Pewe Ho Ntol Lotoy o e Peeto Pe Uety Be 8 P. R. C Att T e eet le o to ott - otool welet e. A yte of ott eto ote fo - otool flte te olto e o
5 - Determinants. r r. r r. r r. r s r = + det det det
5 - Detemts Assote wth y sque mtx A thee s ume lle the etemt of A eote A o et A. Oe wy to efe the etemt, ths futo fom the set of ll mtes to the set of el umes, s y the followg thee popetes. All mtes elow
X-Ray Notes, Part III
oll 6 X-y oe 3: Pe X-Ry oe, P III oe Deeo Coe oupu o x-y ye h look lke h: We efe ue of que lhly ffee efo h ue y ovk: Co: C ΔS S Sl o oe Ro: SR S Co o oe Ro: CR ΔS C SR Pevouly, we ee he SR fo ye hv pxel
Fredholm Type Integral Equations with Aleph-Function. and General Polynomials
Iteto Mthetc Fou Vo. 8 3 o. 989-999 HIKI Ltd.-h.co Fedho Te Iteg uto th eh-fucto d Gee Poo u J K.J. o Ittute o Mgeet tude & eech Mu Id u5@g.co Kt e K.J. o Ittute o Mgeet tude & eech Mu Id dehuh_3@hoo.co
African Journal of Science and Technology (AJST) Science and Engineering Series Vol. 4, No. 2, pp GENERALISED DELETION DESIGNS
Af Joul of See Tehology (AJST) See Egeeg See Vol. 4, No.,. 7-79 GENERALISED DELETION DESIGNS Mhel Ku Gh Joh Wylff Ohbo Dee of Mhe, Uvey of Nob, P. O. Bo 3097, Nob, Key ABSTRACT:- I h e yel gle ele fol
CHAPTER 7 SOLVING FUZZY LINEAR FRACTIONAL PROGRAMMING PROBLEM BY USING TAYLOR'S SERIES METHOD
67 CHAPTER 7 SOLVING FUZZY LINEAR FRACTIONAL PROGRAMMING PROBLEM BY USING TAYLOR'S SERIES METHOD 7. INTRODUCTION The eso mers the setors le fl ororte lg routo lg mretg me seleto uversty lg stuet mssos
ON THE EXTENSION OF WEAK ARMENDARIZ RINGS RELATIVE TO A MONOID
wwweo/voue/vo9iue/ijas_9 9f ON THE EXTENSION OF WEAK AENDAIZ INGS ELATIVE TO A ONOID Eye A & Ayou Eoy Dee of e Nowe No Uvey Lzou 77 C Dee of e Uvey of Kou Ou Su E-: eye76@o; you975@yooo ABSTACT Fo oo we
«A first lesson on Mathematical Induction»
Bcgou ifotio: «A fist lesso o Mtheticl Iuctio» Mtheticl iuctio is topic i H level Mthetics It is useful i Mtheticl copetitios t ll levels It hs bee coo sight tht stuets c out the poof b theticl iuctio,
Studying the Problems of Multiple Integrals with Maple Chii-Huei Yu
Itetol Joul of Resech (IJR) e-issn: 2348-6848, - ISSN: 2348-795X Volume 3, Issue 5, Mch 26 Avlble t htt://tetoljoulofesechog Studyg the Poblems of Multle Itegls wth Mle Ch-Hue Yu Detmet of Ifomto Techology,
TWO INTERFACIAL COLLINEAR GRIFFITH CRACKS IN THERMO- ELASTIC COMPOSITE MEDIA
WO INERFIL OLLINER GRIFFIH RS IN HERMO- ELSI OMOSIE MEDI h m MISHR S DS * Deme o Mheml See I Ie o eholog BHU V-5 I he oee o he le o he e e o eeg o o olle Gh e he ee o he wo ohoo mel e e e emee el. he olem
Chapter 1 Vector Spaces
Chpter Vetor pes - Vetor pes Ler Comtos Vetor spe V V s set over fel F f V F! + V. Eg. R s vetor spe. For R we hek -4=-4-4R -7=-7-7R et. Eg. how tht the set of ll polomls PF wth oeffets from F s vetor
Certain Expansion Formulae Involving a Basic Analogue of Fox s H-Function
vlle t htt:vu.edu l. l. Mth. ISSN: 93-9466 Vol. 3 Iue Jue 8. 8 36 Pevouly Vol. 3 No. lcto d led Mthetc: Itetol Joul M Cet Exo Foule Ivolvg c logue o Fox -Fucto S.. Puoht etet o c-scece Mthetc College o
2. Elementary Linear Algebra Problems
. Eleety e lge Pole. BS: B e lge Suoute (Pog pge wth PCK) Su of veto opoet:. Coputto y f- poe: () () () (3) N 3 4 5 3 6 4 7 8 Full y tee Depth te tep log()n Veto updte the f- poe wth N : ) ( ) ( ) ( )
Exercise # 2.1 3, 7, , 3, , -9, 1, Solution: natural numbers are 3, , -9, 1, 2.5, 3, , , -9, 1, 2 2.5, 3, , -9, 1, , -9, 1, 2.
Chter Chter Syste of Rel uers Tertg Del frto: The del frto whh Gve fte uers of dgts ts del rt s lled tertg del frto. Reurrg ( o-tertg )Del frto: The del frto (No tertg) whh soe dgts re reeted g d g the
Analele Universităţii din Oradea, Fascicula: Protecţia Mediului, Vol. XIII, 2008
Alele Uverstăţ d Orde Fsul: Proteţ Medulu Vol. XIII 00 THEORETICAL AND COMPARATIVE STUDY REGARDING THE MECHANICS DISPLASCEMENTS UNDER THE STATIC LOADINGS FOR THE SQUARE PLATE MADE BY WOOD REFUSE AND MASSIF
CBSE SAMPLE PAPER SOLUTIONS CLASS-XII MATHS SET-2 CBSE , ˆj. cos. SECTION A 1. Given that a 2iˆ ˆj. We need to find
BSE SMLE ER SOLUTONS LSS-X MTHS SET- BSE SETON Gv tht d W d to fd 7 7 Hc, 7 7 7 Lt, W ow tht Thus, osd th vcto quto of th pl z - + z = - + z = Thus th ts quto of th pl s - + z = Lt d th dstc tw th pot,,
Handout on. Crystal Symmetries and Energy Bands
dou o Csl s d g Bds I hs lu ou wll l: Th loshp bw ss d g bds h bs of sp-ob ouplg Th loshp bw ss d g bds h ps of sp-ob ouplg C 7 pg 9 Fh Coll Uvs d g Bds gll hs oh Th sl pol ss ddo o h l slo s: Fo pl h
COMP 465: Data Mining More on PageRank
COMP 465: Dt Mnng Moe on PgeRnk Sldes Adpted Fo: www.ds.og (Mnng Mssve Dtsets) Powe Iteton: Set = 1/ 1: = 2: = Goto 1 Exple: d 1/3 1/3 5/12 9/24 6/15 = 1/3 3/6 1/3 11/24 6/15 1/3 1/6 3/12 1/6 3/15 Iteton
Chapter Linear Regression
Chpte 6.3 Le Regesso Afte edg ths chpte, ou should be ble to. defe egesso,. use sevel mmzg of esdul cte to choose the ght cteo, 3. deve the costts of le egesso model bsed o lest sques method cteo,. use
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
A New Batch FHE Scheme over the Integers
A New Bth FHE Shee oe the Iteges Kw W Sog K Cho U Shoo of Mthets K I Sg Uesty yogyg Deot eoe s e of Koe Astt The FHE (fy hoooh eyto) shees [7 3] sed o the odfed AGCD oe (osefee AGCD oe) e ee to t tts ese
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
Kummer Beta -Weibull Geometric Distribution. A New Generalization of Beta -Weibull Geometric Distribution
ttol Jol of Ss: Bs Al Rsh JSBAR SSN 37-453 Pt & Ol htt://gss.og/.h?joljolofbsaal ---------------------------------------------------------------------------------------------------------------------------
Lecture 3 summary. C4 Lecture 3 - Jim Libby 1
Lecue su Fes of efeece Ivce ude sfoos oo of H wve fuco: d-fucos Eple: e e - µ µ - Agul oeu s oo geeo Eule gles Geec slos cosevo lws d Noehe s heoe C4 Lecue - Lbb Fes of efeece Cosde fe of efeece O whch
SUBSEQUENCE CHARACTERIZAT ION OF UNIFORM STATISTICAL CONVERGENCE OF DOUBLE SEQUENCE
Reseach ad Coucatos atheatcs ad atheatcal ceces Vol 9 Issue 7 Pages 37-5 IN 39-6939 Publshed Ole o Novebe 9 7 7 Jyot cadec Pess htt//yotacadecessog UBEQUENCE CHRCTERIZT ION OF UNIFOR TTITIC CONVERGENCE
Bayesian Estimation of the parameters of the Weibull-Weibull Length-Biased mixture distributions using time censored data
Bys Eso of h s of h Wull-Wull gh-bs xu suos usg so S. A. Sh N Bouss I.S.S. Co Uvsy I.N.P.S. Algs Uvsy shsh@yhoo.o ou005@yhoo.o As I hs h s of h Wull-Wull lgh s xu suos s usg h Gs slg hqu u y I sog sh.
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
Problem Set 4 Solutions
4 Eoom Altos of Gme Theory TA: Youg wg /08/0 - Ato se: A A { B, } S Prolem Set 4 Solutos - Tye Se: T { α }, T { β, β} Se Plyer hs o rte formto, we model ths so tht her tye tke oly oe lue Plyer kows tht
LIPSCHITZ ESTIMATES FOR MULTILINEAR COMMUTATOR OF MARCINKIEWICZ OPERATOR
Reseh d ouiios i heis d hei Siees Vo. Issue Pges -46 ISSN 9-699 Puished Oie o Deee 7 Joi Adei Pess h://oideiess.e IPSHITZ ESTIATES FOR UTIINEAR OUTATOR OF ARINKIEWIZ OPERATOR DAZHAO HEN Dee o Siee d Ioio
Dual-Matrix Approach for Solving the Transportation Problem
Itertol Jourl of Mthets Tres Tehology- Volue Nuer Jue 05 ul-mtr Aroh for Solvg the Trsortto Prole Vy Shr r Chr Bhus Shr ertet of Mthets, BBM College r, Jeh, (MU), INIA E-Prl, SS College Jeh, (MU), INIA
Lecture 9-3/8/10-14 Spatial Description and Transformation
Letue 9-8- tl Deton nd nfomton Homewo No. Due 9. Fme ngement onl. Do not lulte...8..7.8 Otonl et edt hot oof tht = - Homewo No. egned due 9 tud eton.-.. olve oblem:.....7.8. ee lde 6 7. e Mtlb on. f oble.
Prerna Tower, Road No 2, Contractors Area, Bistupur, Jamshedpur , Tel (0657) ,
R Pen Towe Rod No Conttos Ae Bistupu Jmshedpu 8 Tel (67)89 www.penlsses.om IIT JEE themtis Ppe II PART III ATHEATICS SECTION I (Totl ks : ) (Single Coet Answe Type) This setion ontins 8 multiple hoie questions.
Physics 232 Exam II Mar. 28, 2005
Phi 3 M. 8, 5 So. Se # Ne. A piee o gl, ide o eio.5, h hi oig o oil o i. The oil h ide o eio.4.d hike o. Fo wh welegh, i he iile egio, do ou ge o eleio? The ol phe dieee i gie δ Tol δ PhDieee δ i,il δ
Technical Appendix for Inventory Management for an Assembly System with Product or Component Returns, DeCroix and Zipkin, Management Science 2005.
Techc Appedx fo Iveoy geme fo Assemy Sysem wh Poduc o Compoe eus ecox d Zp geme Scece 2005 Lemm µ µ s c Poof If J d µ > µ he ˆ 0 µ µ µ µ µ µ µ µ Sm gumes essh he esu f µ ˆ > µ > µ > µ o K ˆ If J he so
SOME REMARKS ON HORIZONTAL, SLANT, PARABOLIC AND POLYNOMIAL ASYMPTOTE
D I D A C T I C S O F A T H E A T I C S No (4) 3 SOE REARKS ON HORIZONTAL, SLANT, PARABOLIC AND POLYNOIAL ASYPTOTE Tdeusz Jszk Abstct I the techg o clculus, we cosde hozotl d slt symptote I ths ppe the
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
A Deterministic Model for Channel Capacity with Utility
CAPTER 6 A Detestc Model fo Chel Cct wth tlt 6. todcto Chel cct s tl oeto ssocted wth elble cocto d defed s the hghest te t whch foto c be set ove the chel wth btl sll obblt of eo. Chel codg theoes d the
On the k-lucas Numbers of Arithmetic Indexes
Alied Mthetics 0 3 0-06 htt://d.doi.og/0.436/.0.307 Published Olie Octobe 0 (htt://www.scirp.og/oul/) O the -ucs Nubes of Aithetic Idees Segio lco Detet of Mthetics d Istitute fo Alied Micoelectoics (IUMA)
The linear system. The problem: solve
The ler syste The prole: solve Suppose A s vertle, the there ests uue soluto How to effetly opute the soluto uerlly??? A A A evew of dret ethods Guss elto wth pvotg Meory ost: O^ Coputtol ost: O^ C oly
YEAR VSA (1 Mark) SA (4 Marks) LA (6 Marks) Total Marks
VECTOR ALGEBRA D Weghtge 7 Ms SYLLABUS: VECTOR ALGEBRA Vetos sls, mgtue eto of veto Deto oses eto tos of veto Tpes of vetos (equl, ut, eo, pllel olle vetos, posto veto of pot, egtve of veto, ompoets of
On Some Hadamard-Type Inequalıtıes for Convex Functıons
Aville t htt://vuedu/ Al Al Mth ISSN: 93-9466 Vol 9, Issue June 4, 388-4 Alictions nd Alied Mthetics: An Intentionl Jounl AAM On Soe Hdd-Tye Inequlıtıes o, Convex Functıons M Ein Özdei Detent o Mthetics
--- Deceased Information. A1ry't (Ay't olll n5. F\ease turn page ) lslamic Community Center of Tempe. Please print all information clearly.
A1y't (Ay't lll l uty ete ee lese t ll t lely. Deese t st e Mle e s t\e Lee De Bth Age Dte Seultv t\ube h :;;"' :;...".::.."'t ' ' ue uetttte ube use Deth Heste Als ( y); he t vle hve lll be use t etg
Professor Wei Zhu. 1. Sampling from the Normal Population
AMS570 Pofesso We Zhu. Samplg fom the Nomal Populato *Example: We wsh to estmate the dstbuto of heghts of adult US male. It s beleved that the heght of adult US male follows a omal dstbuto N(, ) Def. Smple
Generalisation on the Zeros of a Family of Complex Polynomials
Ieol Joul of hemcs esech. ISSN 976-584 Volume 6 Numbe 4. 93-97 Ieol esech Publco House h://www.house.com Geelso o he Zeos of Fmly of Comlex Polyomls Aee sgh Neh d S.K.Shu Deme of hemcs Lgys Uvesy Fdbd-
4.1 Schrödinger Equation in Spherical Coordinates
Phs 34 Quu Mehs D 9 9 Mo./ Wed./ Thus /3 F./4 Mo., /7 Tues. / Wed., /9 F., /3 4.. -. Shodge Sphe: Sepo & gu (Q9.) 4..-.3 Shodge Sphe: gu & d(q9.) Copuo: Sphe Shodge s 4. Hdoge o (Q9.) 4.3 gu Moeu 4.4.-.
Municipality of Central Elgin User Fee Committee Agenda
ult f etl Elg Use Fee ttee ge Pge 11:.. ttee R # ll t e slsue f Peu Iteest the Geel tue Theef -6 7-18 t f utes t f the utes f eetgs te etebe, 14, Ju 7, 15. suss Ites Reve f the 14 Use Fee heule e Busess
this is the indefinite integral Since integration is the reverse of differentiation we can check the previous by [ ]
![this is the indefinite integral Since integration is the reverse of differentiation we can check the previous by [ ]](/thumbs/78/76749440.jpg "this is the indefinite integral Since integration is the reverse of differentiation we can check the previous by [ ]")
Atervtves The Itegrl Atervtves Ojectve: Use efte tegrl otto for tervtves. Use sc tegrto rules to f tervtves. Aother mportt questo clculus s gve ervtve f the fucto tht t cme from. Ths s the process kow
Flux-linkage equations for 7-winding representation (similar to eq in text)
lux-lkge eutos fo 7-wg epesetto (sl to e.. text The oe tes e efe s follows: Stto-stto tes: Stto-oto tes: oto-stto tes: =s+os =os =os =-[s+os(+] =os =os(- =-[s+os(+5] =s =os(- =s =-[s+os(+] =os =s+os(-
On Fractional Operational Calculus pertaining to the product of H- functions
nenonl eh ounl of Enneen n ehnolo RE e-ssn: 2395-56 Volume: 2 ue: 3 une-25 wwwene -SSN: 2395-72 On Fonl Oeonl Clulu enn o he ou of - funon D VBL Chu, C A 2 Demen of hem, Unve of Rhn, u-3255, n E-ml : vl@hooom
Chapter #2 EEE State Space Analysis and Controller Design
Chpte EEE8- Chpte # EEE8- Stte Spce Al d Cotolle Deg Itodcto to tte pce Obevblt/Cotollblt Modle ede: D D Go - d.go@cl.c.k /4 Chpte EEE8-. Itodcto Ae tht we hve th ode te: f, ', '',.... Ve dffclt to td
A Review of Dynamic Models Used in Simulation of Gear Transmissions
ANALELE UNIVERSITĂłII ETIMIE MURGU REŞIłA ANUL XXI NR. ISSN 5-797 Zol-Ios Ko Io-ol Mulu A Rvw o ls Us Sulo o G Tsssos Th vsgo o lv s lu gg g olg l us o sov sg u o pps g svl s oug o h ps. Th pupos o h ols
RAKE Receiver with Adaptive Interference Cancellers for a DS-CDMA System in Multipath Fading Channels
AKE v wh Apv f Cs fo DS-CDMA Ss Muph Fg Chs JooHu Y Su M EEE JHog M EEE Shoo of E Egg Sou o Uvs Sh-og Gw-gu Sou 5-74 Ko E-: ohu@su As hs pp pv AKE v wh vs og s popos fo DS-CDMA ss uph fg hs h popos pv
= y and Normed Linear Spaces
304-50 LINER SYSTEMS Lectue 8: Solutos to = ad Nomed Lea Spaces 73 Fdg N To fd N, we eed to chaacteze all solutos to = 0 Recall that ow opeatos peseve N, so that = 0 = 0 We ca solve = 0 ecusvel backwads
Theory of Finsler spaces with ( λβ, ) Metric
Theoy of Fsle sces wth ( λβ ) Metc Dhed Thu Kll Multle us Thuv Uvesty Kll DhdhNel E-l: dhedthuc@lco ABTRAT The of ths e s to toduce d study the cocet of ( ) theoes hve ee woout fo ( ) etc whee (x)y s oe
148 CIVIL ENGINEERING
STRUTUR NYSS fluee es fo Bems d Tusses fluee le sows te vto of effet (eto, se d momet ems, foe tuss) used movg ut lod oss te stutue. fluee le s used to deteme te posto of movele set of lods tt uses te
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
are positive, and the pair A, B is controllable. The uncertainty in is introduced to model control failures.
Lectue 4 8. MRAC Desg fo Affe--Cotol MIMO Systes I ths secto, we cosde MRAC desg fo a class of ult-ut-ult-outut (MIMO) olea systes, whose lat dyacs ae lealy aaetezed, the ucetates satsfy the so-called
Nonlocal Boundary Value Problem for Nonlinear Impulsive q k Symmetric Integrodifference Equation
OSR ol o Mec OSR-M e-ssn: 78-578 -SSN: 9-765X Vole e Ve M - A 7 PP 95- wwwojolog Nolocl Bo Vle Poble o Nole lve - Sec egoeece Eo Log Ceg Ceg Ho * Yeg He ee o Mec Yb Uve Yj PR C Abc: A oe ole lve egoeece
Noise in electronic components.
No lto opot5098, JDS No lto opot Th PN juto Th ut thouh a PN juto ha fou opot t: two ffuo ut (hol fo th paa to th aa a lto th oppot to) a thal at oty ha a (hol fo th aa to th paa a lto th oppot to, laka
ELECTROPHORESIS IN STRUCTURED COLLOIDS
ELKIN 4 ELECTROPHORESIS IN STRUCTURE COLLOIS José M. Médez A. Cvestv Mexo I ollboto wt O. Aló-Wess ULA d J. J. Beel-Mstett Cvestv. V µ E; µ 6πη ε ζ ; ζ 3 ε ζ ζ 4 THE GENERATION OF ONE PARTICLE EFFECTIVE
CBSE , ˆj. cos CBSE_2015_SET-1. SECTION A 1. Given that a 2iˆ ˆj. We need to find. 3. Consider the vector equation of the plane.
CBSE CBSE SET- SECTION. Gv tht d W d to fd 7 7 Hc, 7 7 7. Lt,. W ow tht.. Thus,. Cosd th vcto quto of th pl.. z. - + z = - + z = Thus th Cts quto of th pl s - + z = Lt d th dstc tw th pot,, - to th pl.
such that for 1 From the definition of the k-fibonacci numbers, the firsts of them are presented in Table 1. Table 1: First k-fibonacci numbers F 1
Scholas Joual of Egeeg ad Techology (SJET) Sch. J. Eg. Tech. 0; (C):669-67 Scholas Academc ad Scetfc Publshe (A Iteatoal Publshe fo Academc ad Scetfc Resouces) www.saspublshe.com ISSN -X (Ole) ISSN 7-9
NONDIFFERENTIABLE MATHEMATICAL PROGRAMS. OPTIMALITY AND HIGHER-ORDER DUALITY RESULTS
HE PUBLISHING HOUSE PROCEEDINGS OF HE ROMANIAN ACADEMY, See A, OF HE ROMANIAN ACADEMY Volue 9, Nube 3/8,. NONDIFFERENIABLE MAHEMAICAL PROGRAMS. OPIMALIY AND HIGHER-ORDER DUALIY RESULS Vale PREDA Uvety
Useful R-norm Information Measure and its Properties
IOS Jorl of Eletros Coto Eeer (IOS-JECE) e-issn: 7-34- ISSN: 7-735Vole Isse (No - De 03) PP 5-57 DS oo Keert Uyy DKSr 3 Jyee Uersty of Eeer Teoloy AB o or 4736 Dstt G MP (I) Astrt : I te reset oto ew sefl
Matrix. Definition 1... a1 ... (i) where a. are real numbers. for i 1, 2,, m and j = 1, 2,, n (iii) A is called a square matrix if m n.
Mtrx Defto () s lled order of m mtrx, umer of rows ( 橫行 ) umer of olums ( 直列 ) m m m where j re rel umers () B j j for,,, m d j =,,, () s lled squre mtrx f m (v) s lled zero mtrx f (v) s lled detty mtrx
BINOMIAL THEOREM SOLUTION. 1. (D) n. = (C 0 + C 1 x +C 2 x C n x n ) (1+ x+ x 2 +.)
BINOMIAL THEOREM SOLUTION. (D) ( + + +... + ) (+ + +.) The coefficiet of + + + +... + fo. Moeove coefficiet of is + + + +... + if >. So. (B)... e!!!! The equied coefficiet coefficiet of i e -.!...!. (A),
Example: Two Stochastic Process u~u[0,1]
![Example: Two Stochastic Process u~u[0,1]](/thumbs/86/93797001.jpg "Example: Two Stochastic Process u~u[0,1]")
Co o Slo o Coco S Sh EE I Gholo h@h. ll Sochc Slo Dc Slo l h PLL c Mo o coco w h o c o Ic o Co B P o Go E A o o Po o Th h h o q o ol o oc o lco q ccc lco l Bc El: Uo Dbo Ucol Sl Ab bo col l G col G col
χ be any function of X and Y then
We have show that whe we ae gve Y g(), the [ ] [ g() ] g() f () Y o all g ()() f d fo dscete case Ths ca be eteded to clude fuctos of ay ube of ado vaables. Fo eaple, suppose ad Y ae.v. wth jot desty fucto,
Optimality of Strategies for Collapsing Expanded Random Variables In a Simple Random Sample Ed Stanek
Optmlt of Strteges for Collpsg Expe Rom Vrles Smple Rom Smple E Stek troucto We revew the propertes of prectors of ler comtos of rom vrles se o rom vrles su-spce of the orgl rom vrles prtculr, we ttempt
ME 501A Seminar in Engineering Analysis Page 1
Fobeius ethod pplied to Bessel s Equtio Octobe, 7 Fobeius ethod pplied to Bessel s Equtio L Cetto Mechicl Egieeig 5B Sei i Egieeig lsis Octobe, 7 Outlie Review idte Review lst lectue Powe seies solutios/fobeius
FIBONACCI-LIKE SEQUENCE ASSOCIATED WITH K-PELL, K-PELL-LUCAS AND MODIFIED K-PELL SEQUENCES
Joual of Appled Matheatcs ad Coputatoal Mechacs 7, 6(), 59-7 www.ac.pcz.pl p-issn 99-9965 DOI:.75/jac.7..3 e-issn 353-588 FIBONACCI-LIKE SEQUENCE ASSOCIATED WITH K-PELL, K-PELL-LUCAS AND MODIFIED K-PELL
Convergence tests for the cluster DFT calculations
Covgc ss o h clus DF clculos. Covgc wh spc o bss s. s clculos o bss s covgc hv b po usg h PBE ucol o 7 os gg h-b. A s o h Guss bss ss wh csg s usss hs b us clug h -G -G** - ++G(p). A l sc o. Å h c bw h
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
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
G8-11 Congruence Rules
G8-11 ogruee Rules If two polgos re ogruet, ou ple the oe o top of the other so tht the th etl. The verties tht th re lled orrespodig verties. The gles tht th re lled orrespodig gles. The sides tht th
-Z ONGRE::IONAL ACTION ON FY 1987 SUPPLEMENTAL 1/1
-Z-433 6 --OGRE::OA ATO O FY 987 SUPPEMETA / APPR)PRATO RfQUEST PAY AD PROGRAM(U) DE ARTMET OF DEES AS O' D 9J8,:A:SF ED DEFS! WA-H ODM U 7 / A 25 MRGOPf RESOUTO TEST HART / / AD-A 83 96 (~Go w - %A uj
Empirical equations for electrical parameters of asymmetrical coupled microstrip lines
Epl equons fo elel petes of syel ouple osp lnes I.M. Bsee Eletons eseh Instute El-h steet, Dokk, o, Egypt Abstt: Epl equons e eve fo the self n utul nutne n ptne fo two syel ouple osp lnes. he obne ptne
Asymptotic Dominance Problems. is not constant but for n 0, f ( n) 11. 0, so that for n N f
Asymptotc Domce Prolems Dsply ucto : N R tht s Ο( ) ut s ot costt 0 = 0 The ucto ( ) = > 0 s ot costt ut or 0, ( ) Dee the relto " " o uctos rom N to R y g d oly = Ο( g) Prove tht s relexve d trstve (Recll:
Mathematical Induction (selected questions)
Mtheticl Iductio (selected questios). () Let P() e the propositio : For P(), L.H.S. R.H.S., P() is true. Assue P() is true for soe turl uer, tht is, () For P( ),, y () By the Priciple of Mtheticl Iductio,
THIS PAGE DECLASSIFIED IAW EO 12958
THIS PAGE DECLASSIFIED IAW EO 2958 THIS PAGE DECLASSIFIED IAW EO 2958 THIS PAGE DECLASSIFIED IAW E0 2958 S T T T I R F R S T Exhb e 3 9 ( 66 h Bm dn ) c f o 6 8 b o d o L) B C = 6 h oup C L) TO d 8 f f
E-Companion: Mathematical Proofs
E-omnon: Mthemtcl Poo Poo o emm : Pt DS Sytem y denton o t ey to vey tht t ncee n wth d ncee n We dene } ] : [ { M whee / We let the ttegy et o ech etle n DS e ]} [ ] [ : { M w whee M lge otve nume oth
The formulae in this booklet have been arranged according to the unit in which they are first
Fomule Booklet Fomule Booklet The fomule ths ooklet hve ee ge ccog to the ut whch the e fst touce. Thus cte sttg ut m e eque to use the fomule tht wee touce peceg ut e.g. ctes sttg C mght e epecte to use
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
Complete Classification of BKM Lie Superalgebras Possessing Strictly Imaginary Property
Appled Mthemtcs 4: -5 DOI: 59/m4 Complete Clssfcto of BKM Le Supelges Possessg Stctly Imgy Popety N Sthumoothy K Pydhs Rmu Isttute fo Advced study Mthemtcs Uvesty of Mds Che 6 5 Id Astct I ths ppe complete
Summary: Binomial Expansion...! r. where
Summy: Biomil Epsio 009 M Teo www.techmejcmth-sg.wes.com ) Re-cp of Additiol Mthemtics Biomil Theoem... whee )!!(! () The fomul is ville i MF so studets do ot eed to memoise it. () The fomul pplies oly
Parameter Estimation and Hypothesis Testing of Two Negative Binomial Distribution Population with Missing Data
Avlble ole wwwsceceeccom Physcs Poce 0 475 480 0 Ieol Cofeece o Mecl Physcs Bomecl ee Pmee smo Hyohess es of wo Neve Boml Dsbuo Poulo wh Mss D Zhwe Zho Collee of MhemcsJl Noml UvesyS Ch zhozhwe@6com Absc
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
Spring Term 1 SPaG Mat 4
Spg Tem 1 SPG Mt Cmplete the tle g sux t ech u t mke jectve Nu Ajectve c e C u vete cmms t ths ect speech setece? Hw u cete tht lvel pctue? ske the cuus gl C u wte et ech these hmphe ws? Use ct t help
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
ON JENSEN S AND HERMITE-HADAMARD S INEQUALITY
IJRRAS 7 3 Deemer 203 wwwrressom/volumes/vol7issue3/ijrras_7_3_02 ON JENSEN S AND HERMITE-HADAMARD S INEQUALITY Zlto Pvć & Ver Novosel 2 Mehl Egeerg Fulty Slvos Bro Uversty o Osje Trg Ive Brlć Mžurć 2
I if +5sssi$ E sr. Egglg[[l[aggegr glieiffi*gi I I a. gl$[fli$ilg1li3fi[ Ell F rss. F$EArgi. SEgh*rqr. H uf$:xdx. FsfileE
(tl Sh*q +sss$!! ll ss s ;s$ll s ; B 3 $ Sest -9[*; s$t 1,1 - e^ -" H u$xdx fd $A sfle *9,9* '. s. \^ >X!l P s H 2.ue ^ O - HS 1- -l ( l[[l[e lff* l$[fl$l1l3f[ U, -.1 $tse;es s TD T' ' t B $*l$ \l - 1
FRACTIONAL MELLIN INTEGRAL TRANSFORM IN (0, 1/a)
Ieol Jol o Se Reeh Pblo Volme Ie 5 y ISSN 5-5 FRACTIONAL ELLIN INTEGRAL TRANSFOR IN / S.. Kh R..Pe* J.N.Slke** Deme o hem hh Aemy o Egeeg Al-45 Pe I oble No.: 98576F No.: -785759 Eml-mkh@gml.om Deme o
Fairing of Parametric Quintic Splines
ISSN 46-69 Eglad UK Joual of Ifomato ad omputg Scece Vol No 6 pp -8 Fag of Paametc Qutc Sples Yuau Wag Shagha Isttute of Spots Shagha 48 ha School of Mathematcal Scece Fuda Uvesty Shagha 4 ha { P t )}
1 4 6 is symmetric 3 SPECIAL MATRICES 3.1 SYMMETRIC MATRICES. Defn: A matrix A is symmetric if and only if A = A, i.e., a ij =a ji i, j. Example 3.1.
SPECIAL MATRICES SYMMETRIC MATRICES Def: A mtr A s symmetr f d oly f A A, e,, Emple A s symmetr Def: A mtr A s skew symmetr f d oly f A A, e,, Emple A s skew symmetr Remrks: If A s symmetr or skew symmetr,
On Almost Increasing Sequences For Generalized Absolute Summability
Joul of Applied Mthetic & Bioifotic, ol., o., 0, 43-50 ISSN: 79-660 (pit), 79-6939 (olie) Itetiol Scietific Pe, 0 O Alot Iceig Sequece Fo Geelized Abolute Subility W.. Suli Abtct A geel eult coceig bolute