MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES. Institute for Mathematical Research, Universiti Putra Malaysia, UPM Serdang, Selangor, Malaysia

Size: px
Start display at page:

Download "MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES. Institute for Mathematical Research, Universiti Putra Malaysia, UPM Serdang, Selangor, Malaysia"

Transcription

1 Malaysan Journal of Mahemacal Scences 9(2): (2015) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal homeage: h://ensemumedumy/journal A Mehod for Deermnng -Adc Orders of Facorals 1* Rafka Zulkal, 1 Kamel Arffn Mohd Aan and 1,2 S Hasana Saar 1 Insue for Mahemacal Research, Unvers Pura Malaysa, UPM Serdang, Selangor, Malaysa 2 Dearmen of Mahemacs, Faculy of Scence, Unvers Pura Malaysa, UPM Serdang, Selangor, Malaysa E-mal: rafkazulkal@gmalcom, kamel@umedumy and shas@umedumy *Corresondng auhor ABSTRACT In hs aer, wh a rme he adc sze of n! where n s a osve neger s deermned for n 0 and n 0 The dscusson egns wh he deermnaon of he adc szes of facoral funcons!, q! and q! wh, 0 and q a rme dfferen from I s found ha! 1 1 wh 0 Resuls are hen used o oan he exlc form of adc szes of n from works of earler auhors I s also found ha he adc ers of n C r n! n r! r! s gven y ln ln C where n and r wh 0 Keyws: Facoral funcons, adc szes

2 Rafka Zulkal, Kamel Arffn Mohd Aan & S Hasana Saar 1 INTRODUCTION In hs aer, we resen a mehod for deermnng adc ers of n!, for any ove neger n We use he noaon x, where s a rme and x s any raonal numer o denoe he hghes ower of dvdng x We refer o x as he adc er or he adc sze of x I follows ha, for wo raonal numers of x and y, x xy x y, x y and y x y mn x, y By convenon x f x 0 The noaon x wll as usual denoe he greaes neger funcon Wh n! where n general r!n r! n r, n C r wll denoe he quoen x! x x 1 x Lengyel, 2003 dscussed he er of lacunary sums of nomal k n k coeffcen of he form Gn, lk 0 l n where n ndcaes he facoral r n! funcon Hs sudy nvolves negers n of he form n r!n r! where s an odd rme and 0 Adelerg, 1996 examned he adc n ers of n! and where n r and esalshed some new congruence r relaons assocaed wh adc neger er Bernoull numers He aled he relaons o rove he rreducly roery of ceran Bernoull olynomals wh ers ha are dvsle y Wagsaff, 1996 dscussed he Aurfeullan facorzaons and he erod of he Bell numers modulo a rme He showed ha exonenal negers 1 s he mnmum erod modulo of he Bell Malaysan Journal of Mahemacal Scences

3 A Mehod for Deermnng -Adc Orders of Facorals Kolz, 1977 deermned he adc szes of n! where n can e exressed as a numer n ase rme numer n he form 2 s n Sn n a0 a1 a2 as ha s n! wh Sn a 1 he summaon of he coeffcen n n for 0 a 1 Berend, 1997 saed ha here exs nfnely many neger osve n such ha n!!! 0 mod 2 1 n 1 k n where 1, 2,, k are rme facorsn n! n ascendng er Laer, Yong, 2003 mroved he resul oaned y Sander, 2001 and showed ha here exs nal values of n for any rme facorzaon of n! Suose k s any neger and 0,1 for 1, 2,, k I s shown ha here exs nfnely many osve neger n wh n! mod 2, n! mod 2,, n! mod k , Yong and We roved ha f q s rme and l, any osve negers hen lq! l q! l! q q q Based on he works of Kolz, 1997 and Mohd Aan and Loxon, 1986, Saar and Mohd Aan, 2002 examned he coeffcens of lnear aral dervave olynomals f and f assocaed wh he quadrac olynomal 2 2, x y f x y ax xy cy dx ey m and gave he esmae of he adc szes of her common zeros n erms of he adc ers of he coeffcens Laer n 2007, hey examned he coeffcens of a olynomal o arrve a he adc esmae of common zeros of k In h 6 degree f and n erms of he adc ers of he coeffcens n he domnan erms of f x, y For a olynomal n he nomal form, he coeffcens are exressle n erms of he facorals Tha s, n n n n n n! f x, y ax y C ax y where C Such cases n!! 0 necessae a mehod o deermne he adc ers of he facorals n n C We egn our dscusson wh Secon 20 for deermnng he adc ers of! and q! where, q rmes wh q and 0 We also derve a formula for adc ers of q! wh, 0 n hs secon In he susequen Secon 30, we resen a mehod for deermnng x f y Malaysan Journal of Mahemacal Scences 279

4 Rafka Zulkal, Kamel Arffn Mohd Aan & S Hasana Saar k he adc szes of n! where n k wh k 0 for n 0 and n 0 A he end of hs secon, we dscuss he fndngs of revous researchers regardng he deermnaon of adc szes for any facorals In he followng Secon 40, we resen a mehod for deermnng he adc szes of he facoral funcon n C where n and r wh 0 Ths s wh a vew o deermne he adc szes of coeffcens of he erms n he exanson of he olynomal f x, y ax y where n s osve n he cases n 2 -ADIC ORDERS OF FACTORIAL FUNCTIONS Le n e a osve neger, and q rmes We wll frs consder and n q where qand, 0 In hs secon, we wll deermne he adc szes of! and q! followed y he adc szes of q! wh, Adc Orders of α! and β q! We egn our dscusson y nroducng a lemma for deermnng he numer of facors k n! where 0 k 1 such ha k where a non-negave neger as follows Lemma 211 Suose s a rme, 0, 1 k 1 and Then, here exs such ha k 1 facors r k n! 1 1 Proof Le! k Then! k Le k 0 1 k0 e a facor n! k0 such ha k where 280 Malaysan Journal of Mahemacal Scences

5 A Mehod for Deermnng -Adc Orders of Facorals The numer of such facors s he same as he numer of k such ha k snce clearly k Now, k when k wh 0 for some neger Hence,, k, I follows ha Now, 1 k mles ha The numer of such ha 1 k, 1 1 Tha s 1 and Euler oen funcon, 1 1 1, 1, s gven y he 1 1 k Our asseron follows 1 Hence, here exs values of k, 1 k 1 such ha Corollary 211 If s a rme, 0, 1 k 1 and 0 1, hen 1 1! Proof Le! k k Then! k k k1 k1 Le e a non-negave neger By Lemma 211, for each here exs 1 1 facors k where 1 k 1 n! such ha k Therefore,! Malaysan Journal of Mahemacal Scences 281 1

6 Rafka Zulkal, Kamel Arffn Mohd Aan & S Hasana Saar Tha s 0 1 1! 1 Ths corollary wll e aled for he roof of he followng heorem: Theorem 211 Le e a rme and 0 Then 1! 1 Proof By Corollary 211, 1! k1 k2 1 2 k k On smlfyng we oan! Malaysan Journal of Mahemacal Scences

7 A Mehod for Deermnng -Adc Orders of Facorals Nex, we nvesgae he adc szes of! where q s a rme and q and 0 A frs, we nroduce he followng lemma ha wll e used for he nex heorem For a osve neger n we deermne he numer of negers m where 0 m n, whose adc ers are non-vanshng Lemma 212 Le e a rme, n 0 and m s an neger wh n 0 mn Then here exs negers m such ha m 0 Proof Snce m s an neger and m 0, m s of he form m k where k 0 Now, le S m m k, k 1,2,3, e he se of negers m such ha k 0 Then, k gves he numer of negers m such ha m 0 Now, n 0 m n mles ha 0 k n Tha s, 0 k n Snce k s an neger, 0 k n Therefore, S m m k, k 1,2,3,, n n Clearly, here exs elemens n S I follows ha, here exs negers m such ha m 0 We can now aly hs lemma for deermnng he numer of facors q k n q! whose adc ers are non-negave, as follows: Theorem 212 Le q q Then, here exs 1 q ln q k where 0 ln q, q e any rme and q, 0 and 0 k q 1 facors q k n q! such ha Malaysan Journal of Mahemacal Scences 283

8 Rafka Zulkal, Kamel Arffn Mohd Aan & S Hasana Saar q 1 Proof Le q! q k Then q! q k k 0 For each facor q k n! q k when q 1 k 0 q k where 0 Now, q, le q k m where m 0 Thus, he numer of such facors of q! s gven y he numer of negers m such ha m 0 for every We wll deermne hs numer as follows: Clearly, k q m Snce 0 k, we have 0 q m q from q q q whch 0 m Snce m s an neger, 0 m By Lemma 212, q here exs 1 negers m such ha m 0 Therefore, here exs q q 1 negers m such ha m 0 Hence, he numer of facors q k q q n! 1 q such ha q k s gven y q Snce m s an neger wh1 m, we have q Therefore, ln q 0 ln We wll recover he resul of Lemma 211 y leng q n he frs ar of he aove heorem The followng heorem gves he adc er of q! where qand 0 Theorem 213 Le q, e any rme, q and 0 Then q ln q ln q q 1 0! 284 Malaysan Journal of Mahemacal Scences

9 A Mehod for Deermnng -Adc Orders of Facorals q 1 Proof Le q! q k Then q q k k 0 q 1! Le e a non-negave neger By Theorem 212, for each here exs q q 1 facors q k n q! such ha k, where ln q 0 Thus, q ln 22 Adc Orders of k0 ln q ln q q 1 0! q! Theorem 221 gves he adc ers of q! wh, 0 usng he resul from Theorem 213 Theorem 221 Suose and q rme wh ln q ln 1 q q! q q Then q 1 Proof From he defnon, q! q k Ths equaon can e rewren as k0 q,, 0 and 0 1 q 1 q 1 q! q j q wh 0 j0 1, ndcang 1 1 q 1 q q 1 Now, q j q j q! j0 j0 Malaysan Journal of Mahemacal Scences 285

10 Rafka Zulkal, Kamel Arffn Mohd Aan & S Hasana Saar q 1 wh 1 q 1 Therefore, q! q! q Then 1, q 1 1 q 1!! q q q Snce q 1 1, q , q q! q, q 0 for 1 q 1 I follows ha 1 1 q 0 Thus, q q q 1,!! From hs equaon, y relacng we have 1 1 q! q q! where Therefore, q! q q q q q q! q! q Summng he geomerc rogresson, we oan 1 q! q q! 1 1 q! q q! 1 Tha s, 286 Malaysan Journal of Mahemacal Scences

11 A Mehod for Deermnng -Adc Orders of Facorals From Theorem 213, wh q, we have q ln q ln q q 1 0! I follows ha ln q ln 1 q q! q q 3 -ADIC ORDERS OF n! Le e any rme and n a osve neger In hs secon, we resen our man resuls on deermnng he adc szes of n! where n s exressed n s rme ower decomoson of he form k n k wh 0 for 1,2,, k In er o deermne he adc szes of n!, we need o consder he value of n whch can e dvded no wo cases They are negers n such ha n 0 and n 0 31 Adc Orders of n! wh n 0 In hs secon, we dscuss he adc ers of n! wh n 0 as n he followng heorem n1 From he defnon, n! n k Therefore, n! n k k0 n1 k0 Suose n k s a facor n n! wh 0 k n The followng lemma and heorem show ha he adc szes of n! deends on he numer of hese facors The resul of Lemma 212 s used n he roof of Lemma 311 as follows: Lemma 311 Suose s a rme and n e a osve neger wh n 0 and 0 k n Le e a non-negave neger Then, here exs Malaysan Journal of Mahemacal Scences 287

12 Rafka Zulkal, Kamel Arffn Mohd Aan & S Hasana Saar n n 1 ln n 0 ln facors n k n! n such ha n k wh Proof n1 From he defnon, n! n k, n! n k k0 n1 Suose n k s a facor n n! wh k0,1,2,, n 1 and a nonnegave neger Now, n k when Thus, he numer of such facors of n k0 n k m wh m 0 k such ha s gven y he numer of negers m m 0 for each Now, consder he facors n n k m n k n n! k such ha n k Thus where m 0 Snce k 0 and k n m, we have 0 m n n n Snce m s an neger, 0 m, here exs negers m for every k such ha m 0 n From Lemma 212, here exs 1 n n follows ha here exs 1 every negers m such ha m 0 I negers m such ha m 0 for Thus, he numer of such facors n k wh k0,1,2,, n 1 n n! such n k s gven y n n 1 for every ha 288 Malaysan Journal of Mahemacal Scences

13 A Mehod for Deermnng -Adc Orders of Facorals n Snce m s an neger wh 1m, we have n Hence, ln n 0 ln The followng Theorem 311 gves he adc szes of n! for n 0 usng he resul from Lemma 311 Theorem 311 Suose s a rme, n a osve neger wh n 0 and a non-negave neger Then, n! n1 ln n ln n n 1 0 Proof Snce n! n k, we have n! n k k0 n1 k0 n n From Lemma 311, here exs 1 k0,1,2,, n 1 n! follows ha n! n such ha ln n ln 1 0 n k wh facors n k wh ln n 0 ln I n n 32 Adc Orders of n! wh n 0 In hs secon, we resen he case n whch he adc ers of n! s osve The followng heorem gves a resul on he adc szes of such n! y usng resul from Theorem 311 Theorem 321 Le e any rme and n a osve neger such ha n where 0 Then, Malaysan Journal of Mahemacal Scences 289

14 Proof Gven n Clearly, n1 Rafka Zulkal, Kamel Arffn Mohd Aan & S Hasana Saar ln n ln 1 n n n! n n wh 0 Then, n n1 wh n From he defnon, n! n1! n1 k n k0 The roduc on he rgh hand sde of he equaon can e rewren as a roduc of wo facors accng o he adc szes of k, whch are k 0 and k 0 n 1 n Thus, n! n1! n1 j n1 where ha 0 n 1 n 1 j0 1, n1 1 Now, n1 j n1 j n1! j0 j0 n1 1 1 n1 1 Therefore, n1! n1! n1 1, n 1 Then n1! n1! n1 1 1 n1 ndcaes 1 1, 1 1 Tha s, n1! n1 n1! n1 Snce n 11 1,, n1 0 for 1 n1 1 n 11 n1 0 1, I follows ha 290 Malaysan Journal of Mahemacal Scences

15 A Mehod for Deermnng -Adc Orders of Facorals Thus, n! n n! (1) Le e an neger n he range 1 1 Then y relacng y Therefore y (1) and (2), n Equaon (1) we would have 1 1 n! n n! (2) n! n! n n n + + n + + n n 1 n1! n 1 n! 1 1 Hence, n! n n 1! 1 On smlfyng, we oan n! n1 n1! 1 Snce n1 0, from Theorem 311, we have n ln n1 ln n1 n1 1! 1 0 Malaysan Journal of Mahemacal Scences 291

16 Thus, Rafka Zulkal, Kamel Arffn Mohd Aan & S Hasana Saar ln n1 ln 1 n1 n1! n n Leng n 1 ln n ln 1 n n follows ha n! n n, The followng corollary of Theorem 321 shows he adc ers of n! where n wh 0 Corollary 321 Suose n s any osve neger wh rme ower k decomoson n k Le n n 1,2,, k Then, ln n ln 1 n n! n n for k Proof Gven n k wh 0 for 1,2,, k Clearly, n 0 As n he roof of Theorem 321, we oan 1 n! n! n 1 wh n 0 for 1,2,, k Snce n 0, y Theorem 311 we have ln n ln n n! 1 0 n Hence, ln n ln 1 n n! n n wh n n 292 Malaysan Journal of Mahemacal Scences

17 A Mehod for Deermnng -Adc Orders of Facorals Kolz, 1977 showed ha f s any rme and n osve neger where n s exressed as a numer n ase rme of he form 2 s n Sn n a0 a1 a2 as and 0 a 1, hen n! 1, he summaon of he coeffcens of n n, 0 where Sn a comarson, Theorem 311 and Theorem 321 gve n! ln n ln n n for 0 n and 1 0 ln n ln 1 n n n! n for n 0 resecvely Boh formulae gve alernave ways of deermnng n! ased on ceran condons wh he laer whou havng o exress he value of n n ase rme numer By In he revous fndngs of adc szes of arcular facorals y Yong and We, 2007, s shown ha for all rmes q and any osve negers and l, eq lq! l eqq! eq l! The noaon eq xreresens he q adc er of an neger x Based on Theorem 221, y nerchangng rmes, q and ln ln q q 1 1, we oan q q! 1 q 1 0 q q where q and 0 Ths gves he q adc szes of q! for any rme and q Now, we aly our mehod o oan exlc resul for q lq! for any osve neger l We need only o deermne he value of q l! snce ha of q! s readly avalale from Theorem 211 In q er o evaluae q l!, here are wo cases o consder; hey are he case when q l 0 and q l 0 Malaysan Journal of Mahemacal Scences 293

18 Rafka Zulkal, Kamel Arffn Mohd Aan & S Hasana Saar The followng heorem gves he l 0 usng he resul from Theorem 311 q q adc szes of lq! wh q l and Theorem 322 Suose q s any rme and l, are osve negers wh q l and l 0 Suose e a non-negave neger Then q lnl ln q q 1 l l q! 1 q 1 0 q q lq l Proof Yong and We, 2007 showed ha lq! l q! l! q q q wh 0 From Theorem 211, wh q a rme and 0, we have q 1 q q! q 1 As well as from Theorem 311 wh q l 0, we have Thus, ln l ln q l l q l! 1 0 q q lnl ln q q 1 l l q lq! l 1 q 1 0 q q Now, he nex heorem gves he q l, 0 usng he resul from Theorem 321 q adc szes of lq! wh q l and Theorem 323 wh q l Suose q s any rme and and l are osve negers 294 Malaysan Journal of Mahemacal Scences

19 Suose A Mehod for Deermnng -Adc Orders of Facorals q l wh 0 and s a non-negave neger Then, ln lq ln q q 1 l l q! 1 q q 1 0 q q lq l Proof From he works of Yong and We, 2007, s shown ha q lq! l q q! q l! wh 0 From Theorem 211, wh q rme and 0, we oan q 1 q q! q 1 As well as from Theorem 321, wh lnlq ln q q l q 1 l l q l! l 1 q q 1 0 q q Then, lnlq ln q, 0, we have q 1 q 1 l l q lq! l l 1 q 1 q q 1 0 q q Tha s, ln lq ln q q 1 l l q lq! l 1 q q 1 0 q q 4 -ADIC ORDERS OF n C r Le n, r e negers wh n r In hs secon, we wll dscuss he n n! adc szes of Cr, for he case n and r Clearly r! n r! snce n r Now, C!!! Therefore, C!!! Malaysan Journal of Mahemacal Scences 295

20 By Theorem 211, we have Rafka Zulkal, Kamel Arffn Mohd Aan & S Hasana Saar 1 1 C 1 1!! 1 (3) The value of! s deermned y he followng Theorem 411 Frs, we have he followng asseron: Lemma 411 Suose s a rme, 0 and 0 k Then, here exs facors k n ln! such ha k where 0 ln 1 Proof Le! k 1 k0 Then, k! k0 Le k e a facor n! where 0 Now, such ha k k when k m where m 0 Thus, he numer of such facors n! s gven y he numer of negers m such ha m 0 for every Now, k m Tha s, k 0,1,2,, 1 m k wh 296 Malaysan Journal of Mahemacal Scences

21 Thus, 0 m A Mehod for Deermnng -Adc Orders of Facorals Snce m s neger hen 0 m By Lemma 212, here exs m negers m such ha Therefore, here exs m negers m such ha Hence, he numer of facors of k n! such ha k s gven y 1 1 Snce m s an neger and 1 m, we have Therefore, ln 0 ln Theorem 411 Le e any rme, 0, hen Proof ln ln 1 1! 0 1 1! k k k0 k0 Malaysan Journal of Mahemacal Scences 297

22 Rafka Zulkal, Kamel Arffn Mohd Aan & S Hasana Saar From Lemma 411, here exs k n! ln 0 ln Thus, 1 1 such ha k ln ln facors, where 1 1! 0 We nex deermne he value of C as follows: Theorem 412 Le e a rme, 0, hen ln ln C Proof The roof follows from equaon (3) and Theorem CONCLUSION In hs aer, we have resened a mehod for deermnng he adc szes of n! where n s a osve neger and s a rme The resuls oaned are n exlc forms and he mehod of oanng hem offers an alernave way o fndng n! As resened n hs aer, he mehod does no requre he neger n o e exressed n ase as s usually done I also enales one o oan more exlc resuls of adc szes of lq! where l s an neger, q a rme and 0 To llusrae alcaon of resuls oaned n hs aer, adc szes of n C where n r wh 0 are deermned Ths mehod s exendale o r and 298 Malaysan Journal of Mahemacal Scences

23 A Mehod for Deermnng -Adc Orders of Facorals deermnng n C r for any osve n and r The adc szes of oher exressons conanng facoral facors may also e found y alyng he resuls n hs aer ACKNOWLEDGEMENT We would lke o acknowledge wh hanks he fnancal suor for our research from Fundamenal Research Gran Scheme of Malaysa REFERENCES Adullah, A R (1991) The four on Exlc Decouled Grou (EDG) mehod: A fas Posson solver Inernaonal Journal of Comuer Mahemacs 38: Adelerg, A (1996) Congruences of -Adc Order Bernoull Numers Journal of Numer Theory 59: Berend, D (1997) On he Pary of Exonens n he Facorzaon of n! Journal of Numer Theory 64: Kolz, N (1977) -Adc Numers, -Adc Analyss and Zea Funcons NewYork: Srnger-Verlag Lengyel, T (2003) On he Order of Lacunary Sums of Bnomal Coeffcens Elecronc Journal of Comnaoral Numer Theory 31:10-12 Mohd Aan, K A and J H Loxon (1986) Newon Polyhedra and Soluons of Congruences In Loxon, JH and Van der Pooren, A(ed) Dohanne Analyss Camrdge : Camrdge Unversy Press Sander, J W (2001) On he Pary of Exonens n he Prme Facorzaon of Facorals Journal of Numer Theory 90: Saar, S H, and K A Mohd Aan (2002) Penganggaran Kekardnalan Se Penyelesaan Persamaan Kongruen Jurnal Teknolog 36(C) : Malaysan Journal of Mahemacal Scences 299

24 Rafka Zulkal, Kamel Arffn Mohd Aan & S Hasana Saar Saar, S H, and K A Mohd Aan (2007a) Penganggaran Saz -Adc Pensfar Seunya Teran Seara Polnomal Berdarjah Enam Sans Malaysana 36(C) : Saar, S H, and K A Mohd Aan (2007) A Mehod of Esmang -Adc Szes of Common Zeros of Paral Dervave Polynomals Assocaed Wh an n h Degree Form Malaysan Journal of Mahemacs Scences 1(1) : Wagsaff, Jr S S (1996) Aurfeullan Facorzaons and he Perod of he Bell Numers Modulo a Prme Mahemacs of Comuaon 65: Yong, G C (2003) On he Pary of Exonens n he Sandard Facorzaon of n! Journal of Numer Theory 100: Yong, G C and We, L (2007) On he Prme Power Facorzaon of! n (Par 2) Journal of Numer Theory 122: Malaysan Journal of Mahemacal Scences

Lecture 18: The Laplace Transform (See Sections and 14.7 in Boas)

Lecture 18: The Laplace Transform (See Sections and 14.7 in Boas) Lecure 8: The Lalace Transform (See Secons 88- and 47 n Boas) Recall ha our bg-cure goal s he analyss of he dfferenal equaon, ax bx cx F, where we emloy varous exansons for he drvng funcon F deendng on

More information

A New Generalized Gronwall-Bellman Type Inequality

A New Generalized Gronwall-Bellman Type Inequality 22 Inernaonal Conference on Image, Vson and Comung (ICIVC 22) IPCSIT vol. 5 (22) (22) IACSIT Press, Sngaore DOI:.7763/IPCSIT.22.V5.46 A New Generalzed Gronwall-Bellman Tye Ineualy Qnghua Feng School of

More information

GENERATING CERTAIN QUINTIC IRREDUCIBLE POLYNOMIALS OVER FINITE FIELDS. Youngwoo Ahn and Kitae Kim

GENERATING CERTAIN QUINTIC IRREDUCIBLE POLYNOMIALS OVER FINITE FIELDS. Youngwoo Ahn and Kitae Kim Korean J. Mah. 19 (2011), No. 3, pp. 263 272 GENERATING CERTAIN QUINTIC IRREDUCIBLE POLYNOMIALS OVER FINITE FIELDS Youngwoo Ahn and Kae Km Absrac. In he paper [1], an explc correspondence beween ceran

More information

Solution in semi infinite diffusion couples (error function analysis)

Solution in semi infinite diffusion couples (error function analysis) Soluon n sem nfne dffuson couples (error funcon analyss) Le us consder now he sem nfne dffuson couple of wo blocks wh concenraon of and I means ha, n a A- bnary sysem, s bondng beween wo blocks made of

More information

PHYS 705: Classical Mechanics. Canonical Transformation

PHYS 705: Classical Mechanics. Canonical Transformation PHYS 705: Classcal Mechancs Canoncal Transformaon Canoncal Varables and Hamlonan Formalsm As we have seen, n he Hamlonan Formulaon of Mechancs,, are ndeenden varables n hase sace on eual foong The Hamlon

More information

Online Supplement for Dynamic Multi-Technology. Production-Inventory Problem with Emissions Trading

Online Supplement for Dynamic Multi-Technology. Production-Inventory Problem with Emissions Trading Onlne Supplemen for Dynamc Mul-Technology Producon-Invenory Problem wh Emssons Tradng by We Zhang Zhongsheng Hua Yu Xa and Baofeng Huo Proof of Lemma For any ( qr ) Θ s easy o verfy ha he lnear programmng

More information

FI 3103 Quantum Physics

FI 3103 Quantum Physics /9/4 FI 33 Quanum Physcs Aleander A. Iskandar Physcs of Magnesm and Phooncs Research Grou Insu Teknolog Bandung Basc Conces n Quanum Physcs Probably and Eecaon Value Hesenberg Uncerany Prncle Wave Funcon

More information

On elements with index of the form 2 a 3 b in a parametric family of biquadratic elds

On elements with index of the form 2 a 3 b in a parametric family of biquadratic elds On elemens wh ndex of he form a 3 b n a paramerc famly of bquadrac elds Bora JadrevĆ Absrac In hs paper we gve some resuls abou prmve negral elemens p(c p n he famly of bcyclc bquadrac elds L c = Q ) c;

More information

Cubic Bezier Homotopy Function for Solving Exponential Equations

Cubic Bezier Homotopy Function for Solving Exponential Equations Penerb Journal of Advanced Research n Compung and Applcaons ISSN (onlne: 46-97 Vol. 4, No.. Pages -8, 6 omoopy Funcon for Solvng Eponenal Equaons S. S. Raml *,,. Mohamad Nor,a, N. S. Saharzan,b and M.

More information

Approximate Analytic Solution of (2+1) - Dimensional Zakharov-Kuznetsov(Zk) Equations Using Homotopy

Approximate Analytic Solution of (2+1) - Dimensional Zakharov-Kuznetsov(Zk) Equations Using Homotopy Arcle Inernaonal Journal of Modern Mahemacal Scences, 4, (): - Inernaonal Journal of Modern Mahemacal Scences Journal homepage: www.modernscenfcpress.com/journals/jmms.aspx ISSN: 66-86X Florda, USA Approxmae

More information

Advanced time-series analysis (University of Lund, Economic History Department)

Advanced time-series analysis (University of Lund, Economic History Department) Advanced me-seres analss (Unvers of Lund, Economc Hsor Dearmen) 3 Jan-3 Februar and 6-3 March Lecure 4 Economerc echnues for saonar seres : Unvarae sochasc models wh Box- Jenns mehodolog, smle forecasng

More information

In the complete model, these slopes are ANALYSIS OF VARIANCE FOR THE COMPLETE TWO-WAY MODEL. (! i+1 -! i ) + [(!") i+1,q - [(!

In the complete model, these slopes are ANALYSIS OF VARIANCE FOR THE COMPLETE TWO-WAY MODEL. (! i+1 -! i ) + [(!) i+1,q - [(! ANALYSIS OF VARIANCE FOR THE COMPLETE TWO-WAY MODEL The frs hng o es n wo-way ANOVA: Is here neracon? "No neracon" means: The man effecs model would f. Ths n urn means: In he neracon plo (wh A on he horzonal

More information

HEAT CONDUCTION PROBLEM IN A TWO-LAYERED HOLLOW CYLINDER BY USING THE GREEN S FUNCTION METHOD

HEAT CONDUCTION PROBLEM IN A TWO-LAYERED HOLLOW CYLINDER BY USING THE GREEN S FUNCTION METHOD Journal of Appled Mahemacs and Compuaonal Mechancs 3, (), 45-5 HEAT CONDUCTION PROBLEM IN A TWO-LAYERED HOLLOW CYLINDER BY USING THE GREEN S FUNCTION METHOD Sansław Kukla, Urszula Sedlecka Insue of Mahemacs,

More information

V.Abramov - FURTHER ANALYSIS OF CONFIDENCE INTERVALS FOR LARGE CLIENT/SERVER COMPUTER NETWORKS

V.Abramov - FURTHER ANALYSIS OF CONFIDENCE INTERVALS FOR LARGE CLIENT/SERVER COMPUTER NETWORKS R&RATA # Vol.) 8, March FURTHER AALYSIS OF COFIDECE ITERVALS FOR LARGE CLIET/SERVER COMPUTER ETWORKS Vyacheslav Abramov School of Mahemacal Scences, Monash Unversy, Buldng 8, Level 4, Clayon Campus, Wellngon

More information

Comparison of Differences between Power Means 1

Comparison of Differences between Power Means 1 In. Journal of Mah. Analyss, Vol. 7, 203, no., 5-55 Comparson of Dfferences beween Power Means Chang-An Tan, Guanghua Sh and Fe Zuo College of Mahemacs and Informaon Scence Henan Normal Unversy, 453007,

More information

Appendix H: Rarefaction and extrapolation of Hill numbers for incidence data

Appendix H: Rarefaction and extrapolation of Hill numbers for incidence data Anne Chao Ncholas J Goell C seh lzabeh L ander K Ma Rober K Colwell and Aaron M llson 03 Rarefacon and erapolaon wh ll numbers: a framewor for samplng and esmaon n speces dversy sudes cology Monographs

More information

Inverse Joint Moments of Multivariate. Random Variables

Inverse Joint Moments of Multivariate. Random Variables In J Conem Mah Scences Vol 7 0 no 46 45-5 Inverse Jon Momens of Mulvarae Rom Varables M A Hussan Dearmen of Mahemacal Sascs Insue of Sascal Sudes Research ISSR Caro Unversy Egy Curren address: Kng Saud

More information

CONSISTENT ESTIMATION OF THE NUMBER OF DYNAMIC FACTORS IN A LARGE N AND T PANEL. Detailed Appendix

CONSISTENT ESTIMATION OF THE NUMBER OF DYNAMIC FACTORS IN A LARGE N AND T PANEL. Detailed Appendix COSISE ESIMAIO OF HE UMBER OF DYAMIC FACORS I A LARGE AD PAEL Dealed Aendx July 005 hs verson: May 9, 006 Dane Amengual Dearmen of Economcs, Prnceon Unversy and Mar W Wason* Woodrow Wlson School and Dearmen

More information

( ) () we define the interaction representation by the unitary transformation () = ()

( ) () we define the interaction representation by the unitary transformation () = () Hgher Order Perurbaon Theory Mchael Fowler 3/7/6 The neracon Represenaon Recall ha n he frs par of hs course sequence, we dscussed he chrödnger and Hesenberg represenaons of quanum mechancs here n he chrödnger

More information

Existence and Uniqueness Results for Random Impulsive Integro-Differential Equation

Existence and Uniqueness Results for Random Impulsive Integro-Differential Equation Global Journal of Pure and Appled Mahemacs. ISSN 973-768 Volume 4, Number 6 (8), pp. 89-87 Research Inda Publcaons hp://www.rpublcaon.com Exsence and Unqueness Resuls for Random Impulsve Inegro-Dfferenal

More information

Comb Filters. Comb Filters

Comb Filters. Comb Filters The smple flers dscussed so far are characered eher by a sngle passband and/or a sngle sopband There are applcaons where flers wh mulple passbands and sopbands are requred Thecomb fler s an example of

More information

EP2200 Queuing theory and teletraffic systems. 3rd lecture Markov chains Birth-death process - Poisson process. Viktoria Fodor KTH EES

EP2200 Queuing theory and teletraffic systems. 3rd lecture Markov chains Birth-death process - Poisson process. Viktoria Fodor KTH EES EP Queung heory and eleraffc sysems 3rd lecure Marov chans Brh-deah rocess - Posson rocess Vora Fodor KTH EES Oulne for oday Marov rocesses Connuous-me Marov-chans Grah and marx reresenaon Transen and

More information

. The geometric multiplicity is dim[ker( λi. number of linearly independent eigenvectors associated with this eigenvalue.

. The geometric multiplicity is dim[ker( λi. number of linearly independent eigenvectors associated with this eigenvalue. Lnear Algebra Lecure # Noes We connue wh he dscusson of egenvalues, egenvecors, and dagonalzably of marces We wan o know, n parcular wha condons wll assure ha a marx can be dagonalzed and wha he obsrucons

More information

New M-Estimator Objective Function. in Simultaneous Equations Model. (A Comparative Study)

New M-Estimator Objective Function. in Simultaneous Equations Model. (A Comparative Study) Inernaonal Mahemacal Forum, Vol. 8, 3, no., 7 - HIKARI Ld, www.m-hkar.com hp://dx.do.org/.988/mf.3.3488 New M-Esmaor Objecve Funcon n Smulaneous Equaons Model (A Comparave Sudy) Ahmed H. Youssef Professor

More information

CH.3. COMPATIBILITY EQUATIONS. Continuum Mechanics Course (MMC) - ETSECCPB - UPC

CH.3. COMPATIBILITY EQUATIONS. Continuum Mechanics Course (MMC) - ETSECCPB - UPC CH.3. COMPATIBILITY EQUATIONS Connuum Mechancs Course (MMC) - ETSECCPB - UPC Overvew Compably Condons Compably Equaons of a Poenal Vecor Feld Compably Condons for Infnesmal Srans Inegraon of he Infnesmal

More information

A NEW TECHNIQUE FOR SOLVING THE 1-D BURGERS EQUATION

A NEW TECHNIQUE FOR SOLVING THE 1-D BURGERS EQUATION S19 A NEW TECHNIQUE FOR SOLVING THE 1-D BURGERS EQUATION by Xaojun YANG a,b, Yugu YANG a*, Carlo CATTANI c, and Mngzheng ZHU b a Sae Key Laboraory for Geomechancs and Deep Underground Engneerng, Chna Unversy

More information

[ ] 2. [ ]3 + (Δx i + Δx i 1 ) / 2. Δx i-1 Δx i Δx i+1. TPG4160 Reservoir Simulation 2018 Lecture note 3. page 1 of 5

[ ] 2. [ ]3 + (Δx i + Δx i 1 ) / 2. Δx i-1 Δx i Δx i+1. TPG4160 Reservoir Simulation 2018 Lecture note 3. page 1 of 5 TPG460 Reservor Smulaon 08 page of 5 DISCRETIZATIO OF THE FOW EQUATIOS As we already have seen, fne dfference appromaons of he paral dervaves appearng n he flow equaons may be obaned from Taylor seres

More information

Testing a new idea to solve the P = NP problem with mathematical induction

Testing a new idea to solve the P = NP problem with mathematical induction Tesng a new dea o solve he P = NP problem wh mahemacal nducon Bacground P and NP are wo classes (ses) of languages n Compuer Scence An open problem s wheher P = NP Ths paper ess a new dea o compare he

More information

. The geometric multiplicity is dim[ker( λi. A )], i.e. the number of linearly independent eigenvectors associated with this eigenvalue.

. The geometric multiplicity is dim[ker( λi. A )], i.e. the number of linearly independent eigenvectors associated with this eigenvalue. Mah E-b Lecure #0 Noes We connue wh he dscusson of egenvalues, egenvecors, and dagonalzably of marces We wan o know, n parcular wha condons wll assure ha a marx can be dagonalzed and wha he obsrucons are

More information

Relative controllability of nonlinear systems with delays in control

Relative controllability of nonlinear systems with delays in control Relave conrollably o nonlnear sysems wh delays n conrol Jerzy Klamka Insue o Conrol Engneerng, Slesan Techncal Unversy, 44- Glwce, Poland. phone/ax : 48 32 37227, {jklamka}@a.polsl.glwce.pl Keywor: Conrollably.

More information

Sampling Procedure of the Sum of two Binary Markov Process Realizations

Sampling Procedure of the Sum of two Binary Markov Process Realizations Samplng Procedure of he Sum of wo Bnary Markov Process Realzaons YURY GORITSKIY Dep. of Mahemacal Modelng of Moscow Power Insue (Techncal Unversy), Moscow, RUSSIA, E-mal: gorsky@yandex.ru VLADIMIR KAZAKOV

More information

Keywords: Hedonic regressions; hedonic indexes; consumer price indexes; superlative indexes.

Keywords: Hedonic regressions; hedonic indexes; consumer price indexes; superlative indexes. Hedonc Imuaon versus Tme Dummy Hedonc Indexes Erwn Dewer, Saeed Herav and Mck Slver December 5, 27 (wh a commenary by Jan de Haan) Dscusson Paer 7-7, Dearmen of Economcs, Unversy of Brsh Columba, 997-873

More information

M. Y. Adamu Mathematical Sciences Programme, AbubakarTafawaBalewa University, Bauchi, Nigeria

M. Y. Adamu Mathematical Sciences Programme, AbubakarTafawaBalewa University, Bauchi, Nigeria IOSR Journal of Mahemacs (IOSR-JM e-issn: 78-578, p-issn: 9-765X. Volume 0, Issue 4 Ver. IV (Jul-Aug. 04, PP 40-44 Mulple SolonSoluons for a (+-dmensonalhroa-sasuma shallow waer wave equaon UsngPanlevé-Bӓclund

More information

TSS = SST + SSE An orthogonal partition of the total SS

TSS = SST + SSE An orthogonal partition of the total SS ANOVA: Topc 4. Orhogonal conrass [ST&D p. 183] H 0 : µ 1 = µ =... = µ H 1 : The mean of a leas one reamen group s dfferen To es hs hypohess, a basc ANOVA allocaes he varaon among reamen means (SST) equally

More information

Ordinary Differential Equations in Neuroscience with Matlab examples. Aim 1- Gain understanding of how to set up and solve ODE s

Ordinary Differential Equations in Neuroscience with Matlab examples. Aim 1- Gain understanding of how to set up and solve ODE s Ordnary Dfferenal Equaons n Neuroscence wh Malab eamples. Am - Gan undersandng of how o se up and solve ODE s Am Undersand how o se up an solve a smple eample of he Hebb rule n D Our goal a end of class

More information

Dynamic Regressions with Variables Observed at Different Frequencies

Dynamic Regressions with Variables Observed at Different Frequencies Dynamc Regressons wh Varables Observed a Dfferen Frequences Tlak Abeysnghe and Anhony S. Tay Dearmen of Economcs Naonal Unversy of Sngaore Ken Rdge Crescen Sngaore 96 January Absrac: We consder he roblem

More information

Tight results for Next Fit and Worst Fit with resource augmentation

Tight results for Next Fit and Worst Fit with resource augmentation Tgh resuls for Nex F and Wors F wh resource augmenaon Joan Boyar Leah Epsen Asaf Levn Asrac I s well known ha he wo smple algorhms for he classc n packng prolem, NF and WF oh have an approxmaon rao of

More information

Notes on the stability of dynamic systems and the use of Eigen Values.

Notes on the stability of dynamic systems and the use of Eigen Values. Noes on he sabl of dnamc ssems and he use of Egen Values. Source: Macro II course noes, Dr. Davd Bessler s Tme Seres course noes, zarads (999) Ineremporal Macroeconomcs chaper 4 & Techncal ppend, and Hamlon

More information

ON THE WEAK LIMITS OF SMOOTH MAPS FOR THE DIRICHLET ENERGY BETWEEN MANIFOLDS

ON THE WEAK LIMITS OF SMOOTH MAPS FOR THE DIRICHLET ENERGY BETWEEN MANIFOLDS ON THE WEA LIMITS OF SMOOTH MAPS FOR THE DIRICHLET ENERGY BETWEEN MANIFOLDS FENGBO HANG Absrac. We denfy all he weak sequenal lms of smooh maps n W (M N). In parcular, hs mples a necessary su cen opologcal

More information

Chapter Lagrangian Interpolation

Chapter Lagrangian Interpolation Chaper 5.4 agrangan Inerpolaon Afer readng hs chaper you should be able o:. dere agrangan mehod of nerpolaon. sole problems usng agrangan mehod of nerpolaon and. use agrangan nerpolans o fnd deraes and

More information

On One Analytic Method of. Constructing Program Controls

On One Analytic Method of. Constructing Program Controls Appled Mahemacal Scences, Vol. 9, 05, no. 8, 409-407 HIKARI Ld, www.m-hkar.com hp://dx.do.org/0.988/ams.05.54349 On One Analyc Mehod of Consrucng Program Conrols A. N. Kvko, S. V. Chsyakov and Yu. E. Balyna

More information

THE POLYNOMIAL TENSOR INTERPOLATION

THE POLYNOMIAL TENSOR INTERPOLATION Pease ce hs arce as: Grzegorz Berna, Ana Ceo, The oynoma ensor neroaon, Scenfc Research of he Insue of Mahemacs and Comuer Scence, 28, oume 7, Issue, ages 5-. The webse: h://www.amcm.cz./ Scenfc Research

More information

John Geweke a and Gianni Amisano b a Departments of Economics and Statistics, University of Iowa, USA b European Central Bank, Frankfurt, Germany

John Geweke a and Gianni Amisano b a Departments of Economics and Statistics, University of Iowa, USA b European Central Bank, Frankfurt, Germany Herarchcal Markov Normal Mxure models wh Applcaons o Fnancal Asse Reurns Appendx: Proofs of Theorems and Condonal Poseror Dsrbuons John Geweke a and Gann Amsano b a Deparmens of Economcs and Sascs, Unversy

More information

Variational method to the second-order impulsive partial differential equations with inconstant coefficients (I)

Variational method to the second-order impulsive partial differential equations with inconstant coefficients (I) Avalable onlne a www.scencedrec.com Proceda Engneerng 6 ( 5 4 Inernaonal Worksho on Aomoble, Power and Energy Engneerng Varaonal mehod o he second-order mlsve aral dfferenal eqaons wh nconsan coeffcens

More information

Should Exact Index Numbers have Standard Errors? Theory and Application to Asian Growth

Should Exact Index Numbers have Standard Errors? Theory and Application to Asian Growth Should Exac Index umbers have Sandard Errors? Theory and Applcaon o Asan Growh Rober C. Feensra Marshall B. Rensdorf ovember 003 Proof of Proposon APPEDIX () Frs, we wll derve he convenonal Sao-Vara prce

More information

Online Appendix for. Strategic safety stocks in supply chains with evolving forecasts

Online Appendix for. Strategic safety stocks in supply chains with evolving forecasts Onlne Appendx for Sraegc safey socs n supply chans wh evolvng forecass Tor Schoenmeyr Sephen C. Graves Opsolar, Inc. 332 Hunwood Avenue Hayward, CA 94544 A. P. Sloan School of Managemen Massachuses Insue

More information

PARTIAL QUOTIENTS AND DISTRIBUTION OF SEQUENCES. Department of Mathematics University of California Riverside, CA

PARTIAL QUOTIENTS AND DISTRIBUTION OF SEQUENCES. Department of Mathematics University of California Riverside, CA PARTIAL QUOTIETS AD DISTRIBUTIO OF SEQUECES 1 Me-Chu Chang Deartment of Mathematcs Unversty of Calforna Rversde, CA 92521 mcc@math.ucr.edu Abstract. In ths aer we establsh average bounds on the artal quotents

More information

SOME NOISELESS CODING THEOREMS OF INACCURACY MEASURE OF ORDER α AND TYPE β

SOME NOISELESS CODING THEOREMS OF INACCURACY MEASURE OF ORDER α AND TYPE β SARAJEVO JOURNAL OF MATHEMATICS Vol.3 (15) (2007), 137 143 SOME NOISELESS CODING THEOREMS OF INACCURACY MEASURE OF ORDER α AND TYPE β M. A. K. BAIG AND RAYEES AHMAD DAR Absrac. In hs paper, we propose

More information

On computing differential transform of nonlinear non-autonomous functions and its applications

On computing differential transform of nonlinear non-autonomous functions and its applications On compung dfferenal ransform of nonlnear non-auonomous funcons and s applcaons Essam. R. El-Zahar, and Abdelhalm Ebad Deparmen of Mahemacs, Faculy of Scences and Humanes, Prnce Saam Bn Abdulazz Unversy,

More information

ABSTRACT KEYWORDS. Bonus-malus systems, frequency component, severity component. 1. INTRODUCTION

ABSTRACT KEYWORDS. Bonus-malus systems, frequency component, severity component. 1. INTRODUCTION EERAIED BU-MAU YTEM ITH A FREQUECY AD A EVERITY CMET A IDIVIDUA BAI I AUTMBIE IURACE* BY RAHIM MAHMUDVAD AD HEI HAAI ABTRACT Frangos and Vronos (2001) proposed an opmal bonus-malus sysems wh a frequency

More information

Including the ordinary differential of distance with time as velocity makes a system of ordinary differential equations.

Including the ordinary differential of distance with time as velocity makes a system of ordinary differential equations. Soluons o Ordnary Derenal Equaons An ordnary derenal equaon has only one ndependen varable. A sysem o ordnary derenal equaons consss o several derenal equaons each wh he same ndependen varable. An eample

More information

Performance Analysis for a Network having Standby Redundant Unit with Waiting in Repair

Performance Analysis for a Network having Standby Redundant Unit with Waiting in Repair TECHNI Inernaonal Journal of Compung Scence Communcaon Technologes VOL.5 NO. July 22 (ISSN 974-3375 erformance nalyss for a Nework havng Sby edundan Un wh ang n epar Jendra Sngh 2 abns orwal 2 Deparmen

More information

Density Matrix Description of NMR BCMB/CHEM 8190

Density Matrix Description of NMR BCMB/CHEM 8190 Densy Marx Descrpon of NMR BCMBCHEM 89 Operaors n Marx Noaon If we say wh one bass se, properes vary only because of changes n he coeffcens weghng each bass se funcon x = h< Ix > - hs s how we calculae

More information

January Examinations 2012

January Examinations 2012 Page of 5 EC79 January Examnaons No. of Pages: 5 No. of Quesons: 8 Subjec ECONOMICS (POSTGRADUATE) Tle of Paper EC79 QUANTITATIVE METHODS FOR BUSINESS AND FINANCE Tme Allowed Two Hours ( hours) Insrucons

More information

Lecture 11 SVM cont

Lecture 11 SVM cont Lecure SVM con. 0 008 Wha we have done so far We have esalshed ha we wan o fnd a lnear decson oundary whose margn s he larges We know how o measure he margn of a lnear decson oundary Tha s: he mnmum geomerc

More information

Scattering at an Interface: Oblique Incidence

Scattering at an Interface: Oblique Incidence Course Insrucor Dr. Raymond C. Rumpf Offce: A 337 Phone: (915) 747 6958 E Mal: rcrumpf@uep.edu EE 4347 Appled Elecromagnecs Topc 3g Scaerng a an Inerface: Oblque Incdence Scaerng These Oblque noes may

More information

Density Matrix Description of NMR BCMB/CHEM 8190

Density Matrix Description of NMR BCMB/CHEM 8190 Densy Marx Descrpon of NMR BCMBCHEM 89 Operaors n Marx Noaon Alernae approach o second order specra: ask abou x magnezaon nsead of energes and ranson probables. If we say wh one bass se, properes vary

More information

First-order piecewise-linear dynamic circuits

First-order piecewise-linear dynamic circuits Frs-order pecewse-lnear dynamc crcus. Fndng he soluon We wll sudy rs-order dynamc crcus composed o a nonlnear resse one-por, ermnaed eher by a lnear capacor or a lnear nducor (see Fg.. Nonlnear resse one-por

More information

3. OVERVIEW OF NUMERICAL METHODS

3. OVERVIEW OF NUMERICAL METHODS 3 OVERVIEW OF NUMERICAL METHODS 3 Inroducory remarks Ths chaper summarzes hose numercal echnques whose knowledge s ndspensable for he undersandng of he dfferen dscree elemen mehods: he Newon-Raphson-mehod,

More information

P R = P 0. The system is shown on the next figure:

P R = P 0. The system is shown on the next figure: TPG460 Reservor Smulaon 08 page of INTRODUCTION TO RESERVOIR SIMULATION Analycal and numercal soluons of smple one-dmensonal, one-phase flow equaons As an nroducon o reservor smulaon, we wll revew he smples

More information

F-Tests and Analysis of Variance (ANOVA) in the Simple Linear Regression Model. 1. Introduction

F-Tests and Analysis of Variance (ANOVA) in the Simple Linear Regression Model. 1. Introduction ECOOMICS 35* -- OTE 9 ECO 35* -- OTE 9 F-Tess and Analyss of Varance (AOVA n he Smple Lnear Regresson Model Inroducon The smple lnear regresson model s gven by he followng populaon regresson equaon, or

More information

FTCS Solution to the Heat Equation

FTCS Solution to the Heat Equation FTCS Soluon o he Hea Equaon ME 448/548 Noes Gerald Reckenwald Porland Sae Unversy Deparmen of Mechancal Engneerng gerry@pdxedu ME 448/548: FTCS Soluon o he Hea Equaon Overvew Use he forward fne d erence

More information

ON THE ADDITION OF UNITS AND NON-UNITS IN FINITE COMMUTATIVE RINGS

ON THE ADDITION OF UNITS AND NON-UNITS IN FINITE COMMUTATIVE RINGS ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 45, Number 6, 2015 ON THE ADDITION OF UNITS AND NON-UNITS IN FINITE COMMUTATIVE RINGS DARIUSH KIANI AND MOHSEN MOLLAHAJIAGHAEI ABSTRACT. Le R be a fne commuave

More information

Robustness Experiments with Two Variance Components

Robustness Experiments with Two Variance Components Naonal Insue of Sandards and Technology (NIST) Informaon Technology Laboraory (ITL) Sascal Engneerng Dvson (SED) Robusness Expermens wh Two Varance Componens by Ana Ivelsse Avlés avles@ns.gov Conference

More information

Some congruences related to harmonic numbers and the terms of the second order sequences

Some congruences related to harmonic numbers and the terms of the second order sequences Mathematca Moravca Vol. 0: 06, 3 37 Some congruences related to harmonc numbers the terms of the second order sequences Neşe Ömür Sbel Koaral Abstract. In ths aer, wth hels of some combnatoral denttes,

More information

On Local Existence and Blow-Up of Solutions for Nonlinear Wave Equations of Higher-Order Kirchhoff Type with Strong Dissipation

On Local Existence and Blow-Up of Solutions for Nonlinear Wave Equations of Higher-Order Kirchhoff Type with Strong Dissipation Inernaonal Journal of Modern Nonlnear Theory and Alcaon 7 6-5 h://wwwscrorg/journal/jna ISSN Onlne: 67-987 ISSN Prn: 67-979 On Local Exsence and Blow-U of Soluons for Nonlnear Wave Euaons of Hgher-Order

More information

CS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 4

CS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 4 CS434a/54a: Paern Recognon Prof. Olga Veksler Lecure 4 Oulne Normal Random Varable Properes Dscrmnan funcons Why Normal Random Varables? Analycally racable Works well when observaon comes form a corruped

More information

Power-sum problem, Bernoulli Numbers and Bernoulli Polynomials.

Power-sum problem, Bernoulli Numbers and Bernoulli Polynomials. Power-sum roblem, Bernoull Numbers and Bernoull Polynomals. Arady M. Alt Defnton 1 Power um Problem Fnd the sum n : 1... n where, n N or, usng sum notaton, n n n closed form. Recurrence for n Exercse Usng

More information

Chapter 6: AC Circuits

Chapter 6: AC Circuits Chaper 6: AC Crcus Chaper 6: Oulne Phasors and he AC Seady Sae AC Crcus A sable, lnear crcu operang n he seady sae wh snusodal excaon (.e., snusodal seady sae. Complee response forced response naural response.

More information

Bayesian Inference of the GARCH model with Rational Errors

Bayesian Inference of the GARCH model with Rational Errors 0 Inernaonal Conference on Economcs, Busness and Markeng Managemen IPEDR vol.9 (0) (0) IACSIT Press, Sngapore Bayesan Inference of he GARCH model wh Raonal Errors Tesuya Takash + and Tng Tng Chen Hroshma

More information

J i-1 i. J i i+1. Numerical integration of the diffusion equation (I) Finite difference method. Spatial Discretization. Internal nodes.

J i-1 i. J i i+1. Numerical integration of the diffusion equation (I) Finite difference method. Spatial Discretization. Internal nodes. umercal negraon of he dffuson equaon (I) Fne dfference mehod. Spaal screaon. Inernal nodes. R L V For hermal conducon le s dscree he spaal doman no small fne spans, =,,: Balance of parcles for an nernal

More information

2/20/2013. EE 101 Midterm 2 Review

2/20/2013. EE 101 Midterm 2 Review //3 EE Mderm eew //3 Volage-mplfer Model The npu ressance s he equalen ressance see when lookng no he npu ermnals of he amplfer. o s he oupu ressance. I causes he oupu olage o decrease as he load ressance

More information

DEEP UNFOLDING FOR MULTICHANNEL SOURCE SEPARATION SUPPLEMENTARY MATERIAL

DEEP UNFOLDING FOR MULTICHANNEL SOURCE SEPARATION SUPPLEMENTARY MATERIAL DEEP UNFOLDING FOR MULTICHANNEL SOURCE SEPARATION SUPPLEMENTARY MATERIAL Sco Wsdom, John Hershey 2, Jonahan Le Roux 2, and Shnj Waanabe 2 Deparmen o Elecrcal Engneerng, Unversy o Washngon, Seale, WA, USA

More information

LOCATION CHOICE OF FIRMS UNDER STACKELBERG INFORMATION ASYMMETRY. Serhij Melnikov 1,2

LOCATION CHOICE OF FIRMS UNDER STACKELBERG INFORMATION ASYMMETRY. Serhij Melnikov 1,2 TRANPORT & OGITI: he Inernaonal Journal Arcle hsory: Receved 8 March 8 Acceed Arl 8 Avalable onlne 5 Arl 8 IN 46-6 Arcle caon nfo: Melnov,., ocaon choce of frms under acelberg nformaon asymmery. Transor

More information

Pavel Azizurovich Rahman Ufa State Petroleum Technological University, Kosmonavtov St., 1, Ufa, Russian Federation

Pavel Azizurovich Rahman Ufa State Petroleum Technological University, Kosmonavtov St., 1, Ufa, Russian Federation VOL., NO. 5, MARCH 8 ISSN 89-668 ARN Journal of Engneerng and Aled Scences 6-8 Asan Research ublshng Nework ARN. All rghs reserved. www.arnjournals.com A CALCULATION METHOD FOR ESTIMATION OF THE MEAN TIME

More information

OP = OO' + Ut + Vn + Wb. Material We Will Cover Today. Computer Vision Lecture 3. Multi-view Geometry I. Amnon Shashua

OP = OO' + Ut + Vn + Wb. Material We Will Cover Today. Computer Vision Lecture 3. Multi-view Geometry I. Amnon Shashua Comuer Vson 27 Lecure 3 Mul-vew Geomer I Amnon Shashua Maeral We Wll Cover oa he srucure of 3D->2D rojecon mar omograh Marces A rmer on rojecve geomer of he lane Eolar Geomer an Funamenal Mar ebrew Unvers

More information

Linear Response Theory: The connection between QFT and experiments

Linear Response Theory: The connection between QFT and experiments Phys540.nb 39 3 Lnear Response Theory: The connecon beween QFT and expermens 3.1. Basc conceps and deas Q: ow do we measure he conducvy of a meal? A: we frs nroduce a weak elecrc feld E, and hen measure

More information

Mohammad H. Al-Towaiq a & Hasan K. Al-Bzoor a a Department of Mathematics and Statistics, Jordan University of

Mohammad H. Al-Towaiq a & Hasan K. Al-Bzoor a a Department of Mathematics and Statistics, Jordan University of Ths arcle was downloaded by: [Jordan Unv. of Scence & Tech] On: 05 Aprl 05, A: 0:4 Publsher: Taylor & Francs Informa Ld Regsered n England and ales Regsered umber: 07954 Regsered offce: Mormer House, 37-4

More information

NATIONAL UNIVERSITY OF SINGAPORE PC5202 ADVANCED STATISTICAL MECHANICS. (Semester II: AY ) Time Allowed: 2 Hours

NATIONAL UNIVERSITY OF SINGAPORE PC5202 ADVANCED STATISTICAL MECHANICS. (Semester II: AY ) Time Allowed: 2 Hours NATONAL UNVERSTY OF SNGAPORE PC5 ADVANCED STATSTCAL MECHANCS (Semeser : AY 1-13) Tme Allowed: Hours NSTRUCTONS TO CANDDATES 1. Ths examnaon paper conans 5 quesons and comprses 4 prned pages.. Answer all

More information

Method of upper lower solutions for nonlinear system of fractional differential equations and applications

Method of upper lower solutions for nonlinear system of fractional differential equations and applications Malaya Journal of Maemak, Vol. 6, No. 3, 467-472, 218 hps://do.org/1.26637/mjm63/1 Mehod of upper lower soluons for nonlnear sysem of fraconal dfferenal equaons and applcaons D.B. Dhagude1 *, N.B. Jadhav2

More information

Transient Response in Electric Circuits

Transient Response in Electric Circuits Transen esponse n Elecrc rcus The elemen equaon for he branch of he fgure when he source s gven by a generc funcon of me, s v () r d r ds = r Mrs d d r (')d' () V The crcu s descrbed by he opology equaons

More information

The Performance of Optimum Response Surface Methodology Based on MM-Estimator

The Performance of Optimum Response Surface Methodology Based on MM-Estimator The Performance of Opmum Response Surface Mehodology Based on MM-Esmaor Habshah Md, Mohd Shafe Musafa, Anwar Frano Absrac The Ordnary Leas Squares (OLS) mehod s ofen used o esmae he parameers of a second-order

More information

Fuzzy Set Theory in Modeling Uncertainty Data. via Interpolation Rational Bezier Surface Function

Fuzzy Set Theory in Modeling Uncertainty Data. via Interpolation Rational Bezier Surface Function Appled Mahemacal Scences, Vol. 7, 013, no. 45, 9 38 HIKARI Ld, www.m-hkar.com Fuzzy Se Theory n Modelng Uncerany Daa va Inerpolaon Raonal Bezer Surface Funcon Rozam Zakara Deparmen of Mahemacs, Faculy

More information

A New Method for Computing EM Algorithm Parameters in Speaker Identification Using Gaussian Mixture Models

A New Method for Computing EM Algorithm Parameters in Speaker Identification Using Gaussian Mixture Models 0 IACSI Hong Kong Conferences IPCSI vol. 9 (0) (0) IACSI Press, Sngaore A New ehod for Comung E Algorhm Parameers n Seaker Idenfcaon Usng Gaussan xure odels ohsen Bazyar +, Ahmad Keshavarz, and Khaoon

More information

Improvement in Estimating Population Mean using Two Auxiliary Variables in Two-Phase Sampling

Improvement in Estimating Population Mean using Two Auxiliary Variables in Two-Phase Sampling Improvemen n Esmang Populaon Mean usng Two Auxlar Varables n Two-Phase amplng Rajesh ngh Deparmen of ascs, Banaras Hndu Unvers(U.P.), Inda (rsnghsa@ahoo.com) Pankaj Chauhan and Nrmala awan chool of ascs,

More information

Department of Economics University of Toronto

Department of Economics University of Toronto Deparmen of Economcs Unversy of Torono ECO408F M.A. Economercs Lecure Noes on Heeroskedascy Heeroskedascy o Ths lecure nvolves lookng a modfcaons we need o make o deal wh he regresson model when some of

More information

Improvement in Estimating Population Mean using Two Auxiliary Variables in Two-Phase Sampling

Improvement in Estimating Population Mean using Two Auxiliary Variables in Two-Phase Sampling Rajesh ngh Deparmen of ascs, Banaras Hndu Unvers(U.P.), Inda Pankaj Chauhan, Nrmala awan chool of ascs, DAVV, Indore (M.P.), Inda Florenn marandache Deparmen of Mahemacs, Unvers of New Meco, Gallup, UA

More information

arxiv: v2 [math.pr] 2 Nov 2015

arxiv: v2 [math.pr] 2 Nov 2015 Weak and srong momens of l r -norms of log-concave vecors arxv:1501.01649v2 [mah.pr] 2 Nov 2015 Rafa l Laa la and Mara Srzelecka revsed verson Absrac We show ha for 1 and r 1 he -h momen of he l r -norm

More information

Part II CONTINUOUS TIME STOCHASTIC PROCESSES

Part II CONTINUOUS TIME STOCHASTIC PROCESSES Par II CONTINUOUS TIME STOCHASTIC PROCESSES 4 Chaper 4 For an advanced analyss of he properes of he Wener process, see: Revus D and Yor M: Connuous marngales and Brownan Moon Karazas I and Shreve S E:

More information

A Cell Decomposition Approach to Online Evasive Path Planning and the Video Game Ms. Pac-Man

A Cell Decomposition Approach to Online Evasive Path Planning and the Video Game Ms. Pac-Man Cell Decomoson roach o Onlne Evasve Pah Plannng and he Vdeo ame Ms. Pac-Man reg Foderaro Vram Raju Slva Ferrar Laboraory for Inellgen Sysems and Conrols LISC Dearmen of Mechancal Engneerng and Maerals

More information

MODELS OF PRODUCTION RUNS FOR MULTIPLE PRODUCTS IN FLEXIBLE MANUFACTURING SYSTEM

MODELS OF PRODUCTION RUNS FOR MULTIPLE PRODUCTS IN FLEXIBLE MANUFACTURING SYSTEM Yugoslav Journal of Oeraons Research (0), Nuer, 307-34 DOI: 0.98/YJOR0307I MODELS OF PRODUCTION RUNS FOR MULTIPLE PRODUCTS IN FLEXIBLE MANUFACTURING SYSTEM Olver ILIĆ, Mlć RADOVIĆ Faculy of Organzaonal

More information

Survival Analysis and Reliability. A Note on the Mean Residual Life Function of a Parallel System

Survival Analysis and Reliability. A Note on the Mean Residual Life Function of a Parallel System Communcaons n Sascs Theory and Mehods, 34: 475 484, 2005 Copyrgh Taylor & Francs, Inc. ISSN: 0361-0926 prn/1532-415x onlne DOI: 10.1081/STA-200047430 Survval Analyss and Relably A Noe on he Mean Resdual

More information

The Matrix Padé Approximation in Systems of Differential Equations and Partial Differential Equations

The Matrix Padé Approximation in Systems of Differential Equations and Partial Differential Equations C Pesano-Gabno, C Gonz_Lez-Concepcon, MC Gl-Farna The Marx Padé Approxmaon n Sysems of Dfferenal Equaons and Paral Dfferenal Equaons C PESTANO-GABINO, C GONZΑLEZ-CONCEPCION, MC GIL-FARIÑA Deparmen of Appled

More information

A DECOMPOSITION METHOD FOR SOLVING DIFFUSION EQUATIONS VIA LOCAL FRACTIONAL TIME DERIVATIVE

A DECOMPOSITION METHOD FOR SOLVING DIFFUSION EQUATIONS VIA LOCAL FRACTIONAL TIME DERIVATIVE S13 A DECOMPOSITION METHOD FOR SOLVING DIFFUSION EQUATIONS VIA LOCAL FRACTIONAL TIME DERIVATIVE by Hossen JAFARI a,b, Haleh TAJADODI c, and Sarah Jane JOHNSTON a a Deparen of Maheacal Scences, Unversy

More information

Volatility Interpolation

Volatility Interpolation Volaly Inerpolaon Prelmnary Verson March 00 Jesper Andreasen and Bran Huge Danse Mares, Copenhagen wan.daddy@danseban.com brno@danseban.com Elecronc copy avalable a: hp://ssrn.com/absrac=69497 Inro Local

More information

UNIVERSITAT AUTÒNOMA DE BARCELONA MARCH 2017 EXAMINATION

UNIVERSITAT AUTÒNOMA DE BARCELONA MARCH 2017 EXAMINATION INTERNATIONAL TRADE T. J. KEHOE UNIVERSITAT AUTÒNOMA DE BARCELONA MARCH 27 EXAMINATION Please answer wo of he hree quesons. You can consul class noes, workng papers, and arcles whle you are workng on he

More information

EE241 - Spring 2003 Advanced Digital Integrated Circuits

EE241 - Spring 2003 Advanced Digital Integrated Circuits EE4 EE4 - rn 00 Advanced Dal Ineraed rcus Lecure 9 arry-lookahead Adders B. Nkolc, J. Rabaey arry-lookahead Adders Adder rees» Radx of a ree» Mnmum deh rees» arse rees Loc manulaons» onvenonal vs. Ln»

More information

Shear Stress-Slip Model for Steel-CFRP Single-Lap Joints under Thermal Loading

Shear Stress-Slip Model for Steel-CFRP Single-Lap Joints under Thermal Loading Shear Sress-Slp Model for Seel-CFRP Sngle-Lap Jons under Thermal Loadng *Ank Agarwal 1), Eha Hamed 2) and Sephen J Foser 3) 1), 2), 3) Cenre for Infrasrucure Engneerng and Safey, School of Cvl and Envronmenal

More information

Mechanics Physics 151

Mechanics Physics 151 Mechancs Physcs 5 Lecure 9 Hamlonan Equaons of Moon (Chaper 8) Wha We Dd Las Tme Consruced Hamlonan formalsm H ( q, p, ) = q p L( q, q, ) H p = q H q = p H = L Equvalen o Lagrangan formalsm Smpler, bu

More information

Graduate Macroeconomics 2 Problem set 5. - Solutions

Graduate Macroeconomics 2 Problem set 5. - Solutions Graduae Macroeconomcs 2 Problem se. - Soluons Queson 1 To answer hs queson we need he frms frs order condons and he equaon ha deermnes he number of frms n equlbrum. The frms frs order condons are: F K

More information