Analysis of the Expected Number of Bit Comparisons Required by Quickselect
|
|
- Philip Fields
- 5 years ago
- Views:
Transcription
1 Aalyi of the Expected Number of Bit Compario Required by Quickelect Jame Alle Fill Departmet of Applied Mathematic ad Statitic The Joh Hopki Uiverity ad ad Takéhiko Nakama Departmet of Applied Mathematic ad Statitic The Joh Hopki Uiverity ad ABSTRACT Whe algorithm for ortig ad earchig are applied to key that are repreeted a bit trig, we ca quatify the performace of the algorithm ot oly i term of the umber of key compario required by the algorithm but alo i term of the umber of bit compario. Some of the tadard ortig ad earchig algorithm have bee aalyzed with repect to key compario but ot with repect to bit compario. I thi paper, we ivetigate the expected umber of bit compario required by Quickelect alo kow a Fid. We develop exact ad aymptotic formulae for the expected umber of bit compario required to fid the mallet or larget key by Quickelect ad how that the expectatio i aymptotically liear with repect to the umber of key. Similar reult are obtaied for the average cae. For fidig key of arbitrary rak, we derive a exact formula for the expected umber of bit compario that uig ratioal arithmetic require oly fiite ummatio rather tha uch operatio a umerical itegratio ad ue it to compute the expectatio for each target rak. AMS ubect claificatio. Primary 68W4; ecodary 68P, 6C5. Key word ad phrae. Quickelect, Fid, earchig algorithm, aymptotic, average-cae aalyi, key compario, bit compario. Date. March, 9. Reearch for both author upported by NSF grat DMS 464, ad by The Joh Hopki Uiverity Acheo J. Duca Fud for the Advacemet of Reearch i Statitic.
2 INTRODUCTION AND SUMMARY Itroductio ad Summary Whe a algorithm for ortig or earchig i aalyzed, the algorithm i uually regarded either a comparig key pairwie irrepective of the key iteral tructure or a operatig o repreetatio uch a bit trig of key. I the former cae, aalye ofte quatify the performace of the algorithm i term of the umber of key compario required to accomplih the tak; Quickelect alo kow a Fid i a example of thoe algorithm that have bee tudied from thi poit of view. I the latter cae, if key are repreeted a bit trig, the aalye quatify the performace of the algorithm i term of the umber of bit compared util it complete it tak. Digital earch tree, for example, have bee examied from thi perpective. I order to fully quatify the performace of a ortig or earchig algorithm ad eable compario betwee key-baed ad digital algorithm, it i ideal to aalyze the algorithm from both poit of view. However, to date, oly Quickort ha bee aalyzed with both approache; ee Fill ad Jao [3]. Before their tudy, Quickort had bee exteively examied with regard to the umber of key compario performed by the algorithm e.g., Kuth [3], Régier [9], Röler [], Kel ad Szpakowki [], Fill ad Jao [], Neiiger ad Rüchedorf [7], but it had ot bee examied with regard to the umber of bit compario i ortig key repreeted a bit trig. I their tudy, Fill ad Jao aumed that key are idepedetly ad uiformly ditributed over, ad that the key are repreeted a bit trig. [They alo coducted the aalyi for a geeral abolutely cotiuou ditributio over,.] They howed that the expected umber of bit compario required to ort key i aymptotically equivalet to l lg a compared to the lead-order term of the expected umber of key compario, which i aymptotically l. We ue l ad lg to deote atural ad biary logarithm, repectively, ad ue log whe the bae doe ot matter for example, i remaider etimate. I thi paper, we ivetigate the expected umber of bit compario required by Quickelect. Hoare [8] itroduced thi earch algorithm, which i treated i mot textbook o algorithm ad data tructure. Quickelect elect the m-th mallet key we call it the rak-m key from a et of ditict key. The key are typically aumed to be ditict, but the algorithm till work with a mior adutmet eve if they are ot ditict. The algorithm fid the target key i a recurive ad radom fahio. Firt, it elect a pivot uiformly at radom from key. Let k deote the rak of the pivot. If k m, the the algorithm retur the pivot. If k > m, the the algorithm recurively operate o the et of key maller tha the pivot ad retur the rak-m key. Similarly, if k < m, the the algorithm recurively operate o the et of key larger tha the pivot ad retur the k m-th mallet key from the ubet. Although previou tudie e.g., Kuth [], Mahmoud et al. [5], Prodier [8], Grübel ad U. Röler [7], Let ad Mahmoud [4], Mahmoud ad Smythe [6], Devroye [], Hwag ad Tai [9] examied Quickelect with regard to key compario, thi tudy i the firt to aalyze the bit complexity of the algorithm. We uppoe that the algorithm i applied to ditict key that are repreeted a bit trig ad that the algorithm operate o idividual bit i order to fid a target key. We alo aume that the key are uiformly ad idepedetly ditributed i,. For itace, coider applyig Quickelect to fid the mallet key amog three key k, k, ad k 3 whoe biary repreetatio are....,...., ad...., re-
3 INTRODUCTION AND SUMMARY pectively. If the algorithm elect k 3 a a pivot, the it compare each of k ad k to k 3 i order to determie the rak of k 3. Whe k ad k 3 are compared, the algorithm require bit compario to determie that k 3 i maller tha k becaue the two key have the ame firt digit ad differ at the ecod digit. Similarly, whe k ad k 3 are compared, the algorithm require 4 bit compario to determie that k 3 i maller tha k. After thee compario, key k 3 ha bee idetified a mallet. Hece the earch for the mallet key require a total of 6 bit compario reultig from the two key compario. We let µm, deote the expected umber of bit compario required to fid the rak-m key i a file of key by Quickelect. By ymmetry, µm, µ + m,. Firt, we develop exact ad aymptotic formulae for µ, µ,, the expected umber of bit compario required to fid the mallet key by Quickelect, a ummarized i the followig theorem. Theorem.. The expected umber µ, of bit compario required by Quickelect to fid the mallet key i a file of key that are idepedetly ad uiformly ditributed i, ha the followig exact ad aymptotic expreio: µ, H + c l l B + l +. l + O,. H ad B deote harmoic ad Beroulli umber, repectively, ad k c : + k l k k With χ k : πik l expreed a ad γ : Euler cotat..577, the cotat c ca alteratively be c γ 9 l 4 l k Z\{} ζ χ k Γ χ k Γ4 χ k χ k..4 The aymptotic formula how that the expected umber of bit compario i aymptotically liear i with lead-order coefficiet approximately equal to Hece the expected umber of bit compario i aymptotically differet from that of key compario required to fid the mallet key oly by a cotat factor the expectatio for key compario i aymptotically. Detail of the derivatio of the formulae are decribed i Sectio 3. Complex-aalytical method are utilized to obtai the aymptotic formula. with c i the form.4 ad eem to be idipeable for obtaiig aymptotic beyod the lead term. [We remark that, although it ivolve the imagiary umber χ k, the expreio.4 i real becaue the term with idice k ad k are complex cougate.] I Sectio 3. we agai ue complex-aalytical method to reexpre.4 i the form.3. Havig doe all thi we upected that there mut be a purely real-aalytical way to obtai directly the lead-order aymptotic µ, c with c i the form.3. Ideed, there i: See Remark 3.3.
4 PRELIMINARIES 3 I Sectio 4 ad 5 we move o to derive exact ad aymptotic expreio for the expected umber of bit compario for the average cae. We deote thi expectatio by µ m,. I the average cae, the parameter m i µm, i coidered a dicrete uiform radom variable; hece µ m, m µm,. The derived aymptotic formula how that µ m, i alo aymptotically liear i ; ee More detailed reult for µ m, are decribed i Sectio 4. Latly, i Sectio 5, we derive a exact expreio of µm, for each fixed m that i uited for computatio. Our prelimiary exact formula for µm, [how i.8] etail ifiite ummatio ad itegratio. A a reult, it i ot a deirable form for umerically computig the expected umber of bit compario. Hece we etablih aother exact formula that oly require fiite ummatio ad ue it to compute µm, for m,...,,,..., 5. The computatio lead to the followig coecture: i for fixed, µm, icreae i m for m + ad i ymmetric about + ; ad ii for fixed m, µm, icreae i aymptotically liearly. Prelimiarie To ivetigate the bit complexity of Quickelect, we follow the geeral approach developed by Fill ad Jao [3]. Let U,..., U deote the key uiformly ad idepedetly ditributed o,, ad let U i deote the rak-i key. The, for i < aume, P {U i ad U are compared} m + i + m i + if m i if i < m < if m.. To determie the firt probability i., ote that U m,..., U remai i the ame ubet util the firt time that oe of them i choe a a pivot. Therefore, U i ad U are compared if ad oly if the firt pivot choe from U m,..., U i either U i or U. Aalogou argumet etablih the other two cae. For < < t <, it i well kow that the oit deity fuctio of U i ad U i give by f Ui,U, t : i t i t.. i,, i,,
5 3 ANALYSIS OF µ, 4 Clearly, the evet that U i ad U are compared i idepedet of the radom variable U i ad U. Hece, defiig P, t, m, P, t, m, P 3, t, m, m i< i<m< i< m m + f U i,u, t,.3 i + f U i,u, t,.4 m i + f U i,u, t,.5 P, t, m, P, t, m, + P, t, m, + P 3, t, m,.6 [the um i.3.5 are double um over i ad ], ad lettig β, t deote the idex of the firt bit at which the key ad t differ, we ca write the expectatio µm, of the umber of bit compario required to fid the rak-m key i a file of key a µm, β, tp, t, m, dt d.7 k l k l k k l l k l k k + P, t, m, dt d;.8 i thi expreio, ote that k repreet the lat bit at which ad t agree. 3 Aalyi of µ, I Sectio 3., we derive the exact expreio for µ, how i Theorem.. I Sectio 3., we prove the aymptotic reult tated i Theorem.. 3. Exact Computatio of µ, Sice the cotributio of P, t, m, or P 3, t, m, to P, t, m, i zero for m, we have P, t,, P, t,, [ee.4 through.6]. Let x :, y : t, z : t. The P, t,, z x i y i z i,, i,, i< z z η i< i,, i,, z η x + y + η dη z z z x i y i η dη η 3 t η + dη. 3.
6 3 ANALYSIS OF µ, 5 Makig the chage of variable v t η + ad itegratig, ad recallig z t, we fid, after ome calculatio, From.8 ad 3., µ, k + k k + k k + k k + k P, t,, k l k l k t. 3. l l k k l k l k l k l k l l k l k P, t,, dt d l k t dt d l k t [l l k l k ] dt k k {l k [l k ] } k + k k k [l l ]. 3.3 l To further traform 3.3, defie a,r B r r r if r if r if r, 3.4 B r deote the r-th Beroulli umber. Let S, : l l. The S, ee Kuth [3], ad k l [l l ] S k, k l S k, S k+, S k, S k, S k+, r From 3.3 ad 3.5, a,r k r µ, r l a,r k+ r r r a,r r a,r k r r. 3.5 k + k a,r k r r. k r
7 3 ANALYSIS OF µ, 6 Here Hece µ, k + k a,r k r r k r r k + a,r kr r k r a,r r k + kr a,r r. The um r r r r r k r a,r r r + r B r r r + r r B r r r r+ r r+ r r a,r r r. r 3.6 ca be implified a follow: r r r r r r r r r + r r r r r r + r. 3.7 r Pluggig 3.7 ito 3.6 ad recallig B k+ for k, we fially obtai [ µ, + r B r r r + r ] r r r r r + + B H + t, 3.8 H deote the -th harmoic umber ad [ B ] t :. 3.9 The lat equality i 3.8 follow from the eay idetity k k H. k k
8 3 ANALYSIS OF µ, 7 3. Aymptotic Aalyi of µ, I order to obtai a aymptotic expreio for µ,, we aalyze t i The followig lemma provide a exact expreio for t that eaily lead to a aymptotic expreio for µ, : Lemma 3.. For, let u : t + t with t ad v : u + u. Let γ deote Euler cotat..577, ad defie χ k : πik l. The i v H+ + + l γ l Σ, + + Σ : k Z\{} ζ χ k Γ + Γ χ k ; l Γ + 3 χ k ii u H + a H + γ l + + l + + Σ, a : 4 9 Σ : + 7 6γ 8 l l k Z\{} k Z\{} ζ χ k Γ χ k l χ k ζ χ k Γ χ k Γ4 χ k χ k, Γ + Γ + χ k ; iii t H + a [ H + H 7 ] l γ + l H 3 + b Σ, b : Σ : k Z\{} k Z\{} ζ χ k Γ χ k l χ k Γ3 χ k, ζ χ k Γ χ k Γ + l χ k Γ + χ k, ad H deote the -th Harmoic umber of order, i.e., H : i i.
9 3 ANALYSIS OF µ, 8 I thi lemma, u ad v are derived i order to obtai the exact expreio for t i iii. From 3.8, the exact expreio for t alo provide a alterative exact expreio for µ,. Before provig Lemma 3., we complete the proof of Theorem. uig part iii. We kow H l + γ + + O 4, 3. H π O Combiig with 3.8 ad Lemma 3.iii, we obtai a aymptotic expreio for µ, : µ, a l l l + l + O. 3. The term O i 3. ha fluctuatio of mall magitude due to Σ, which i periodic i log with amplitude maller tha.. Thu, a how i Theorem., the aymptotic lope i 3. i c a 8 9 Let S deote the um i c: S : k Z\{} + 7 6γ 9 l 4 l k Z\{} ζ χ k Γ χ k Γ4 χ k χ k k Z\{} ζ χ k Γ χ k Γ4 χ k χ k. 3.3 ζ χ k 3 χ k χ k χ k, 3.4 the formula Γ+x xγx i ued to derive the ecod expreio. Both expreio ivolve the imagiary umber χ k, but S i a real umber. We ivetigate S ad expre it uig oly real fuctio. We have the followig reult: Theorem 3.. Let S be the um defied at 3.4. The ad S : S l S ρ, 3.5 k h k, hm : m l m l m! m m, 3.6 k ρ : 7 6γ 36 l.
10 3 ANALYSIS OF µ, 9 Proof of Theorem 3.. Chooe ad fix < θ <. We how that the itegral J : θ+i θ i ζ 3 d equal πi S o the oe had ad equal πi[ρ + S/ l ] o the other had. Equatig thee two expreio give the deired reult. To get the firt expreio for J, we calculate J k θ+i θ i ζ k 3 d θ+i k ζt kt t + t + t dt. But, for ay poitive iteger m ad ay α >, θ+i θ i ζtm t t + t + t dt α+i α i k θ i ζtm t t + t + t dt πire t [ ζtm t ] t, + t + t which follow from reidue calculu, takig ito accout the cotributio of the imple pole of the itegrad at. Here [ ζtm t ] Re t t + t + t 6 m ad Further, ice α+i α i we have by Melli iverio that ζtm t t + t + t dt α+i α i /m t t + t + t dt. t t + t + 3/4 + / t t + t + /4 t +, equal πi α+i α i x t t + t + t dt fx : 3 4 l x + x 4 x for x ad equal for x. Note that thi require oly α >. So α+i ζtm t α i t + t + t dt πi [ f πi m m l m l m! 3 m ] 4 m πi [hm + 6 ] m, the um i over m or m, ad therefore θ+i θ i ζtm t t dt πihm. + t + t
11 3 ANALYSIS OF µ, Thu we obtai our firt expreio for J. We remark that the erie S coverge geometrically rapidly. To obtai the ecod expreio for J we move the horizotal i.e., real coordiate of the vertical lie of itegratio over from θ to C C i large poitive umber C. By reidue calculu, we fid a deired. [ J πi {Re ζ 3 ] + S } πi ρ + S, l l Uig Theorem 3. it i traightforward to derive the alterative expreio k c + k l 3.7 k for the liear coefficiet c i 3.3. Graber ad Prodiger [6] obtaied a earlier draft of thi maucript ad idepedetly coducted a imilar aalyi of S leadig to 3.7. They alo howed how to compute c efficietly to high preciio ad i particular computed c to 5 decimal place. Remark 3.3. The lead-order aymptotic µ, c with c i the form 3.7 ca alo be obtaied imply uig real-aalytical argumet. Start with.7 with m ad recall that P, t,, P, t,, i give by 3. to ee that µ, t k β, tt [ t + t] d dt. A eay domiated-covergece argumet the how that µ, c with c give i the itegral form Writig β, t c t β, tt d dt. ad t agree i their firt k bit k ad breakig up the double itegral accordig to the firt k bit of t lead to the ummatio form 3.7 of c. We omit the detail. We do ot kow how to obtai aymptotic for µ, beyod the lead term by thi ort of approach. Now we prove Lemma 3.: Proof of Lemma 3.. i Sice [ B + + ] u t + t [ ] B, B [ ]
12 3 ANALYSIS OF µ, it follow that + v u + u B [ ] [ ] B B k+ k k + k + [ k+ ] k k ζ k k k + [ k+ 3.8 ] k ζ! πi + [ + d, 3.9 ] C C i a poitively orieted cloed curve that ecircle the iteger,..., ad doe ot iclude or ecircle ay of the followig poit: + χ k χ k : πik l, k Z; ; ad. Equality 3.8 follow from the fact that the Beroulli umber are extrapolated by the Riema zeta fuctio take at oegative iteger: B k kζ k. [The coefficiet k do ot cocer u ice the Beroulli umber of odd idex greater tha vaih.] Equality 3.9 follow from a direct applicatio of reidue calculu, takig ito accout cotributio of the imple pole at the iteger,...,. Let φ deote the itegrad i 3.9: ζ! φ + [ + ]. We coider a poitively orieted rectagular cotour C l with horizotal ide Im λ l ad Im λ l, λ l : l+π l, l Z +, ad vertical ide Re θ ad Re λ l, < θ <. By elemetary boud o φ alog C l ad the fact that θ+i θ i φ d 3. thi i implicit o page 3 of Flaolet ad Sedgewick [5] ad explicitly proved i the Appedix, oe ca how that lim φ d. l C l Accoutig for reidue due to the pole ecircled by C l, we obtai v + Re [φ] + Re [φ] + Re +χk [φ] k Z\{} H+ + + l γ l Σ, Σ : k Z\{} ζ χ k Γ + Γ χ k. 3. l Γ + 3 χ k
13 3 ANALYSIS OF µ, ii We have u t 3 t t 3 9. Hece, from i, u u + v 9 + Here 9 v + + l 9 H H + l H + γ + + l H + γ + + l γ γ 3 l H + l + + l H Σ Σ Σ. 3.3 H H + H H H + H H H + +, 3.5 we aume 3 for 3.4, but 3.5 hold alo for. I regard to Σ, ote that Σ k Z\{} ζ χ k Γ χ k l χ k [ Γ + Γ + Γ + 3 χ k Γ + χ k ], o that Defie Σ k Z\{} Σ : ζ χ k Γ χ k l χ k k Z\{} ζ χ k Γ χ k l χ k [ Γ + Γ + χ k Γ3 ]. 3.6 Γ4 χ k Γ + Γ + χ k. 3.7 The, combiig 3.3, 3.5, ad 3.6, we obtai u H + a H + l + + γ l + + Σ,
14 4 ANALYSIS OF THE AVERAGE CASE: µ m, 3 a : γ 8 l l k Z\{} ζ χ k Γ χ k Γ4 χ k χ k. 3.8 iii Cloely followig the derivatio of u decribed above, we obtai for t t + u u H + a l 3 H γ + l [ H + H 7 H 3 + H + a l γ + l H 3 + b Σ, 3.9 ] Σ b : Σ : k Z\{} k Z\{} ζ χ k Γ χ k l χ k Γ3 χ k, 3.3 ζ χ k Γ χ k Γ + l χ k Γ + χ k Aalyi of the Average Cae: µ m, 4. Exact Computatio of µ m, Here we coider the parameter m i µm, a a dicrete radom variable with probability ma fuctio P {m i}, i,,...,, ad average over m while the parameter i fixed. Thu, uig the otatio defied i.3 through.7, µ m, m, for l,, 3, µm, β, tp, t, m, dt d m β, t P, t, m, dt d µ m, + µ m, + µ 3 m,, m µ l m, β, t P l, t, m, dt d. 4. m
15 4 ANALYSIS OF THE AVERAGE CASE: µ m, 4 Here µ m, µ 3 m,, ice P 3 t,, m +, P, t, m, by a eay ymmetry argumet we omit, ad o Therefore µ 3 m, β, t P 3, t, m, dt d m β t, P 3 t,, m +, dt d m β, t P, t, m, dt d µ m,. m µ m, µ m, + µ m,, 4. ad we will compute µ m, ad µ m, exactly i Sectio Exact Computatio of µ m, We ue the followig lemma i order to compute µ m, exactly: Lemma 4.. β, t P, t, m, dt d m B + Before provig the lemma, we complete the computatio of µ m,. Note that µ m, β, t µ, + P, t, m, dt d m β, t P, t,, dt d + β, t β, t P, t, m, dt d. m P, t, m, dt d m
16 4 ANALYSIS OF THE AVERAGE CASE: µ m, 5 Therefore, by 3.8 ad Lemma 4., we obtai µ m, B + 3 the ecod equality hold ice B B 3 B , 4.3 +!!!!!! 3 [! +!! ] I Sectio 4.. we combie the expreio for µ m, i 4.3 with a imilar expreio for µ m, to obtai a exact expreio for µ m,. The remaider of thi ectio i devoted to provig Lemma 4.. For thi, the followig expreio for P, t, m, will prove ueful: Lemma 4.. Let m ad let x :, y : t, z : t. The the quatity P, t, m, defied at.3 atifie P, t, m, x ξ + y [Υ m,, ξ, x, y, z Υ m,, ξ, x, y, z + Υ 3 m,, ξ, x, y, z] dξ, 4.4
17 4 ANALYSIS OF THE AVERAGE CASE: µ m, 6 Υ m,, ξ, x, y, z : Υ m,, ξ, x, y, z : Υ 3 m,, ξ, x, y, z : x ξ m mξ + y + z m+, m x ξ m m + zξ + y + z m, m x ξ m z m+. m Proof of Lemma 4.. By..3, P, t, m,! m + i! i!! xi y i z m i<! m i m! x i y i z m + m! i m, i, i! m i<! m i m! x i y i z. m! m + i m, i, i! m i< 4.5 I order to compactly decribe the derivatio of 4.4, we defie the followig idefiite itegratio operator T : T fx : x fξ dξ. We really hould write T fx rather tha T fx, but we would like to ue horthad uch a T x x+ + whe >. The operator T treat it argumet f a a fuctio of x; the other variable ivolved i f amely, y ad z are treated a cotat. The otatio T l will deote the l-th iterate of T. I thi otatio, for m < i, i m! i! x i T m x i m, ad the um i 4.5 equal T m m i< m + m i m, i, x i m y i z. Here m + z z m+ z η m+ dη,
18 4 ANALYSIS OF THE AVERAGE CASE: µ m, 7 o m x i m y i z m + i m, i, m i< z m+ T m m x i m y i η +m dη z i m, i, m i< z m+ T m η +m x + y + η m dη z t m z m+ T m η 3 η + dη 4.6 T m z ote that x + y t. Makig the chage of variable v t η + ad itegratig, we obtai, after ome computatio, z η 3 t η + m dη t m + m From 4.5 ad , [ m + t m+ m + + t ] m +. z z 4.7 P, t, m,! m +! T m t [ mz + t m+ m + zz + t m + z m+ ]. Here 4.8 t [ mz + t m+ m + zz + t m + z m+ ] m+ r t r Υm,, r, z, 4.9 m + m Υm,, r, z : m z m+ r m + z m+ r. 4. r r The, ice t x + y, m+ r t r Υm,, r, z m+ r r r Υm,, r, z x y r. 4.
19 4 ANALYSIS OF THE AVERAGE CASE: µ m, 8 From , P, t, m,! m +! T m r m+ r r r Υm,, r, z x y r m+! r r Υm,, r, z y r T m x m +! m+! r r Υm,, r, z y r x +m m +! + + m. r 4. Becaue of the partial fractio expaio it follow that m + + m m! l r r y r x +m + + m r r y From , r x+m m! m l l m l + l + m m x x l m l m! l l m m x x l m l m! l m! l x! P, t, m, m +!m! x m r x m l m l + l +, r y r ξ dξ ξ l r ξ l ξ + y r dξ x ξ m ξ + y r dξ. 4.3 m+ m+ r x ξ m ξ + y Υm,, r, z x x ξ m ξ + y r dξ Υm,, r, zx ξ m ξ + y r dξ m+ r Υm,, r, zξ + y r dξ. 4.4
20 4 ANALYSIS OF THE AVERAGE CASE: µ m, 9 Here, by 4., m+ r Υm,, r, zξ + y r m+ r m [ m m+ r m + r m + r z m+ r m + m r ξ + y r z m+ r m + m[ξ + y + z m+ z m+ m + ξ + yz m ] m + z[ξ + y + z m z m mξ + yz m ] z m+ r ] ξ + y r m+ r m r ξ + y r z m+ r mξ + y + z m+ m + zξ + y + z m + z m Subtitutio of 4.5 ito 4.4 give the deired 4.4. Proof of Lemma 4.. From Lemma 4., we have P, t, m, m x ξ + y [Υ m,, ξ, x, y, z Υ m,, ξ, x, y, z + Υ 3 m,, ξ, x, y, z] dξ. m Here Υ m,, ξ, x, y, z ξ + y + z m ξ + y + z d dw [ d dw m ] x ξ m w m m wξ+y+z { w [x ξ + w x ξ ] } wξ+y+z 4.6 ξ + y + z { w [x ξ + w x ξ ] + w x ξ + w } wξ+y+z x ξ + ξ + y + z 4.7 ote that x + y + z. Similarly, [ Υ m,, ξ, x, y, z z d dw m ad m m ] x ξ m w m+ m wξ+y+z z d [ x ξ + w x ξ ] z [ x ξ + w dw wξ+y+z ] wξ+y+z z, 4.8 Υ 3 m,, ξ, x, y, z m x ξ m z m+ x ξ + z x ξ. m
21 4 ANALYSIS OF THE AVERAGE CASE: µ m, Hece [Υ m,, ξ, x, y, z Υ m,, ξ, x, y, z + Υ 3 m,, ξ, x, y, z] m ξ + y + x ξ + z. 4.9 Therefore, from 4.6 ad 4.9, we obtai m x P, t, m, ξ + y [ ξ + y + x ξ + z ] dξ x x ξ + y { ξ + y + [ ξ + y] } dξ ξ + y ξ + y dξ x + y y x ξ + y dξ t t. 4. We complete the proof by uig 4. to compute β, t m P, t, m, d dt. We have β, t P, t, m, d dt m t β, t t dt d β, t β, t t dt d t Cloely followig the derivatio how i , oe ca how that β, t t dt d Thu, i order to complete the proof, it remai to how that t β, t dt d B dt d
22 4 ANALYSIS OF THE AVERAGE CASE: µ m, Ideed, we have β, t t k+ k + k k k+ k + k k k k k+ k k+ dt d k k + k v t dt d v dv d k v V k+ [ k+ v] W d dv. 4.4 Here k v V k+ [ k+ v] W { d v if v k+ k v if k+ < v k. 4.5 Thu k v k v V k+ [ k+ v] W From 4.4 ad 4.6, we obtai ad 4.3 i proved. β, t d dv [ k+ ] k v dv + v k v dv k+ k t dt d k + k k+ + k + k + k + +, 4.7 k
23 4 ANALYSIS OF THE AVERAGE CASE: µ m, 4.. Exact Computatio of µ m, ad µ m, The derivatio for obtaiig a computatioally preferable exact expreio for µ m, are etirely aalogou to thoe for µ m, decribed i the previou ectio Sectio 4... Thu we omit detail. A decribed i Sectio 3., P, t, m, i zero for m ad for m, o, from 4., µ m, β, t m P, t, m, dt d. 4.8 Therefore we firt derive a computatioally deirable expreio for Agai, let x :, y : t, z : t. The P, t, m, m m i m< m i + S m,, x, y, z i,, i,, m S m,, x, y, z m P, t, m,. x i y i z m S 3 m,, x, y, z, 4.9 S m,, x, y, z : S m,, x, y, z : S 3 m,, x, y, z : i< m i< i< m x i y i z, i + i,, i,, x i y i z, i + i,, i,, x i y i z. i + i,, i,, Fill ad Jao [3] howed that S m,, x, y, z t. Hece S m,, x, y, z m t. 4.3 Followig the derivatio how i 4.5 through 4., oe ca how that S m,, x, y, z m y x[x + z + y ] 4.3 t {[ t ] + t } t. 4.3
24 4 ANALYSIS OF THE AVERAGE CASE: µ m, 3 To obtai a imilar expreio for m S 3m,, x, y, z, we ote that, lettig m : + m, i : +, : + i, S 3 m,, x, y, z i +,, i,, i x y i z i Thu m i < S + m,, z, y, x. S 3 m,, x, y, z m Ipectig , we fid m m S + m,, z, y, x S m,, z, y, x m S 3 m,, x, y, z t From 4.9, 4.3, 4.3, ad 4.34, P, t, m, m t t t t t + Hece, from 4.8 ad 4.35, µ m, + t t t β, t β, t β, t t dt d t dt d t dt d. 4.36
25 4 ANALYSIS OF THE AVERAGE CASE: µ m, 4 Fill ad Jao [3] howed that β, t t dt d [ ] A careful term-by-term ipectio of the derivatio how i reveal that β, t β, t Combiig , we obtai µ m, 4 t dt d t dt d [ + ] [ ] [ ], [ + ] Fially, we complete the exact computatio of µ m,. From 4., 4.3, ad 4.4, we have µ m, µ m, + µ m, B B [ ] We rewrite or combie ome of the term i 4.4 for the aymptotic aalyi of µ m,
26 4 ANALYSIS OF THE AVERAGE CASE: µ m, 5 decribed i the ext ectio. We defie F :, F : F 3 : F 4 : F 5 : 3 3 B, B [ [ ]. 3 ] [ The ecod, third, fourth, ad fifth term i 4.4 ca be writte a 8 F 4, F, 4 9 F 3, ad 4F 4, repectively. The lat three term i 4.4 ca be combied a follow: [ + ] 4 [ ] 4 [ + ] 3 8 [ ] 8 F 5. Therefore 3 µ m, 8 F + 4 F F 3 4F F Aymptotic Aalyi of µ m, We derive a aymptotic expreio for µ m, how i 4.4. The computatio decribed i thi ectio are aalogou to thoe i Sectio 3.. Hece we merely ketch detail to derive the aymptotic expreio. Firt, we aalyze F. A routie complex-aalytical argumet imilar to but much eaier tha the oe decribed i Sectio 3. how that [ ] F +! Re k k + [, 3 + H + H 5 ] H H 5 l + 4 γ l + γ + + O ],
27 4 ANALYSIS OF THE AVERAGE CASE: µ m, 6 Sice F i equal to t, which i defied at 3.9 ad aalyzed i Sectio 3., we already have a aymptotic expreio for F. Next we derive a aymptotic expreio for F 3 : { } F 3! Re k [ ] k H H + l + γ l + O To obtai a aymptotic expreio for F 4, we cloely follow the approach of Sectio 3.. Let ũ : F 4 + F 4. The [ ] B ũ. 3 Let ṽ : ũ + ũ. The, by computatio imilar to thoe performed for v i Sectio 3., ṽ k ζ k k + k + [ k+3 ] k k 3 { } + ζ! Re k + + [ +3 ] [ ] k { + + ζ! Re 3+χk + + [ +3 ] [ ] Hece k Z\{} ũ ũ + ṽ ξ : k Z\{} 9 H + ã + ξ l [ γ l H + + ζ χ k Γ χ k Γ. l Γ + 3 χ k ] ξ, H H l γ + 4 l +, } ã : ξ : 7 36 l 4 7 γ l k Z\{} k Z\{} ζ χ k Γ χ k Γ l χ k Γ + χ k. ζ χ k Γ χ k l χ k Γ4 χ k,
28 4 ANALYSIS OF THE AVERAGE CASE: µ m, 7 Thu F 4 F 4 + ũ 9 H H + ã l 3 + l γ 8 l ã + b ξ + H 3 + l γ + l 4 l, 4.45 b : ξ : k Z\{} k Z\{} ζ χ k Γ χ k l χ k χ k Γ3 χ k, ζ χ k Γ χ k Γ l χ k χ k Γ + χ k. Therefore F 4 9 l + ã + 9 γ + 8 l + O Fially, we aalyze F 5. By computatio that are etirely aalogou to thoe performed for F, F, ad F 4, { } F 5 +! Re k [ ] 3 k { } + +! Re +χk [ ] 3 k Z\{} k Z\{} 4 H l H l 3 [ l H + H + l l H + l + l ] + Γ χ k Γ + l χ k χ k Γ χ k l l γ l l + l l + O Therefore, from ad , we obtai the followig aymptotic formula for µ m, : µ m, 4 + l ã 4 l l + 4 l l + O The aymptotic lope 4 + l ã i approximately 8.73.
29 5 DERIVATION OF A CLOSED FORMULA FOR µm, 8 5 Derivatio of a Cloed Formula for µm, The exact expreio for µm, obtaied i Sectio [ee.8] ivolve ifiite ummatio ad itegratio. Hece it i ot a preferable form for umerically computig the expectatio. I thi ectio, we etablih aother exact expreio for µm, that oly ivolve fiite ummatio. We alo ue the formula to compute µm, for m,...,,,...,. A decribed i Sectio, it follow from equatio.6.8 that, for q,, 3, µ q m, : µm, µ m, + µ m, + µ 3 m,, 5. k k l l k l k l k l k k + P q, t, m, dt d. 5. The ame techique ca be applied to elimiate the ifiite ummatio ad itegratio from each µ q m,. We decribe the techique for obtaiig a cloed expreio of µ m, i detail. Firt, we traform P, t, m, how i.3 o that we ca elimiate the itegratio i µ m,. Defie C i, : m i < m + i,, i,,, 5.3 m i < i a idicator fuctio that equal if the evet i brace hold ad otherwie. Sice i t i t i i t i u i u i u v t v, u v u v it follow that P, t, m, m i< m i< fm C i, C i, f h i u i u v v f fi h f i f t h f i + u t v+u i u v h + f + h i + f t h C f, h, 5.4 f+ C f, h : f+h+ im f+ i C i, f i + h + f + h i +.
30 5 DERIVATION OF A CLOSED FORMULA FOR µm, 9 Thu, from 5. ad 5.4, we ca elimiate the itegratio i µ m, ad expre it uig polyomial i l: µ m, fm Note that f h C 3 f, h k + k k l C 3 f, h : l h+ l h+ l f+ l f+ kf+h+ [l h+ l h+ ][l f+ l f+ ], + f + C f, h. h h + l h+, f [ f + l f+ ] f Hece [ l h+ l ] [ h+ l ] f+ l f+ f h f + h + which ca be rearraged to C 4 f, h, : f+h + f+h f+h+ h + V f W h Therefore, from , we obtai [ ] f+ h+ l +, C 4 f, h, l, 5.6 f + h + [ ] f+. µ m, fm fm f h f h C 3 f, h f+h+ k + k C 5 f, h, k l k + k f+h+ kf+h+ kf+h+ k C 4 f, h, l l, l
31 6 DISCUSSION 3 C 5 f, h, : C 3 f, h C 4 f, h,. Here, a decribed i Sectio 3., k l a,r k r, l r a,r i defied by 3.4. Now defie C 6 f, h,, r : a,r C 5 f, h,. The µ m, C 7 a : fm fm f h f h f+h+ f+h+ a,r C 5 f, h, k + kf+h++r r k C 6 f, h,, r[ f+h++r ] r C 7 a a, 5.7 a fm f hα f+h+ β C 6 f, h,, a + f + h +, i which α : a f ad β : f + h + a. The procedure decribed above ca be applied to derive aalogou exact formulae for µ m, ad µ 3 m,. I order to derive the aalogou exact formula for µ m,, oe eed oly tart the derivatio by chagig the idicator fuctio i C i, [ee 5.3] to i < m < ad follow each tep of the procedure; for µ 3 m,, tart the derivatio by chagig the idicator fuctio to i < m. Uig the cloed exact formulae of µ m,, µ m,, ad µ 3 m,, we computed µm, for, 3,..., ad m,,...,. Figure how the reult, which ugget the followig: i for fixed, µm, icreae i m for m + ad i ymmetric about + ; ii for fixed m, µm, icreae i ; iii max m µm, i aymptotically liear i. 6 Dicuio Our ivetigatio of the bit complexity of Quickelect revealed that the expected umber of bit compario required by Quickelect to fid the mallet or larget key from a et of key i aymptotically liear i with the aymptotic lope approximately equal to Hece aymptotically it differ from the expected umber of key compario to achieve the ame tak oly by a cotat factor. The expectatio for key compario i aymptotically ; ee Kuth [] ad Mahmoud et al. [5]. Thi reult i rather cotrative to the
32 6 DISCUSSION 3 Expectatio of bit compario 8 μm, m Figure : Expected umber of bit compario for Quickelect. The cloed formulae for µ m,, µ m,, ad µ 3 m, were ued to compute µm, for,,..., repreet the umber of key ad m,,..., m repreet the rak of the target key.
33 7 APPENDIX 3 Quickort cae i which ee Fill ad Jao [3] the expected umber of bit compario i aymptotically l lg a the expected umber of key compario i aymptotically l. Our aalyi alo howed that the expected umber of bit compario for the average cae remai aymptotically liear i with the lead-order coefficiet approximately equal to Agai, the expected umber i aymptotically differet from that of key compario for the average cae oly by a cotat factor. The expected umber of key compario for the average cae i aymptotically 3; ee Mahmoud et al. [5]. Although we have yet to etablih a formula aalogou to 3.8 ad 4.4 for the expected umber of bit compario to fid the m-th key for fixed m, we etablihed a exact expreio that oly require fiite ummatio ad ued it to obtai the reult how i Figure. However, the formula remai complex. Writte a a igle expreio, µm, i a eve-fold um of rather elemetary term with each um havig order term i the wort cae; i thi ee, the ruig time of the algorithm for computig µm, i of order 7. The expreio for µm, doe ot allow u to derive a aymptotic formula for it or to prove the three coecture decribed at the ed of Sectio 5. The ituatio i ubtatially better for the expected umber of key compario to fid the m- th key from a et of key; Kuth [] howed that the expectatio ca be writte a [ H m + H m + 3 mh + m ]. I thi paper, we coidered idepedet ad uiformly ditributed key i,. I thi cae, each bit i bit trig i with probability.5. Buildig o the preet work ad that of Fill ad Jao [3], much more geeral key-ditributio are treated by Vallée et al. []. Their geeralizatio further elucidate the complexity of Quickelect ad other algorithm. Ackowledgmet. We thak Philippe Flaolet, Svate Jao, ad Helmut Prodiger for helpful dicuio. 7 Appedix I order to prove 3., it uffice to how that, for ay poitive iteger m, θ+i θ i ζ m d + ote that ad < θ <. Lettig t :, it i thu ufficiet to how that J : ++θ+i ++θ i Uig the reidue theorem, we obtai [ ] J πi k ζ k m k k! + k! + m + +! k ζt m t dt tt + [t + + ]. +i i ζt m t dt tt + [t + + ] ; 7.
34 REFERENCES 33 The i the ecod term here could ut a well be ay real umber exceedig. Here k ζ k m k k! + k! +! + k k B k+ m k + k +! + k! k B k m k k! + k!. Therefore k ζ k m k k! + k! + k m + +! [ + k m + m +! k m +! B k +! k! + k! m+ k + m+ + k + +! m k ] k m + ; 7. for the ecod equality, ee Kuth [] Exercie O the other had, Flaolet et al. [4] howed that +i i ζt m t dt tt + [t + + ] Thu it follow from that J. Referece πi +! m k k m [] L. Devroye. O the probablitic wort-cae time of Fid. Algorithmica, 3:9 33,. [] J. A. Fill ad S. Jao. Quickort aymptotic. Joural of Algorithm, 44:4 8,. [3] J. A. Fill ad S. Jao. The umber of bit compario ued by Quickort: A averagecae aalyi. Proceedig of the ACM-SIAM Sympoium o Dicrete Algorithm, page 93 3, 4. [4] P. Flaolet, P. Graber, P. Kirchehofer, H. Prodiger, ad R. F. Tichy. Melli traform ad aymptotic: digital um. Theoretical Computer Sciece, 3:9 34, 994. [5] P. Flaolet ad R. Sedgewick. Melli traform ad aymptotic: Fiite differece ad Rice itegral. Theoretical Computer Sciece, 44: 4, 995. [6] P. J. Graber ad H. Prodiger. O a cotat ariig i the aalyi of bit compario i Quickelect. Quaetioe Mathematicae, 3:33 36, 8. [7] R. Grübel ad U. Röler. Aymptotic ditributio theory for Hoare electio algorithm. Advace i Applied Probability, 8:5 69, 996. [8] C. R. Hoare. Fid algorithm 65. Commuicatio of the ACM, 4:3 3, 96.
35 REFERENCES 34 [9] H. Hwag ad T. Tai. Quickelect ad the Dickma fuctio. Combiatoric, Probability ad Computig, :353 37,. [] C. Kel ad W. Szpakowki. Quickort algorithm agai reviited. Dicrete Mathematic ad Theoretical Computer Sciece, 3:43 64, 999. [] D. E. Kuth. Mathematical aalyi of algorithm. I Iformatio Proceig 7 Proceedig of IFIP Cogre, Lublaa, 97, page 9 7. North-Hollad, Amterdam, 97. [] D. E. Kuth. The Art of Computer Programmig. Volume : Fudametal Algorithm. Addio-Weley, Readig, Maachuett, 998. [3] D. E. Kuth. The Art of Computer Programmig. Volume 3: Sortig ad Searchig. Addio-Weley, Readig, Maachuett, 998. [4] J. Let ad H. M. Mahmoud. Average-cae aalyi of multiple Quickelect: A algorithm for fidig order tatitic. Statitic ad Probability Letter, 8:99 3, 996. [5] H. M. Mahmoud, R. Modarre, ad R. T. Smythe. Aalyi of Quickelect: A algorithm for order tatitic. RAIRO Iformatique Théorique et Applicatio, 9:55 76, 995. [6] H. M. Mahmoud ad R. T. Smythe. Probabilitic aalyi of multiple Quickelect. Algorithmica, : , 998. [7] R. Neiiger ad L. Rüchedorf. Rate of covergece for Quickelect. Joural of Algorithm, 44:5 6,. [8] H. Prodiger. Multiple Quickelect Hoare Fid algorithm for everal elemet. Iformatio Proceig Letter, 56:3 9, 995. [9] M. Régier. A limitig ditributio of Quickort. RAIRO Iformatique Théorique et Applicatio, 3: , 989. [] U. Röler. A limit theorem for Quickort. RAIRO Iformatique Théorique et Applicatio, 5:85, 99. [] B. Vallée, J. Clémet, J. A. Fill, ad P. Flaolet. The umber of ymbol compario i Quickort ad Quickelect. Preprit, 9.
Analysis of the Expected Number of Bit Comparisons Required by Quickselect
Aalysis of the Expected Number of Bit Comparisos Required by Quickselect James Alle Fill Takéhiko Nakama Abstract Whe algorithms for sortig ad searchig are applied to keys that are represeted as bit strigs,
More informationAnalysis of Execution Costs for QuickSelect. Takéhiko Nakama
Aalysis of Executio Costs for QuickSelect by Takéhiko Nakama A dissertatio submitted to The Johs Hopkis Uiversity i coformity with the requiremets for the degree of Doctor of Philosophy. Baltimore, Marylad
More informationa 1 = 1 a a a a n n s f() s = Σ log a 1 + a a n log n sup log a n+1 + a n+2 + a n+3 log n sup () s = an /n s s = + t i
0 Dirichlet Serie & Logarithmic Power Serie. Defiitio & Theorem Defiitio.. (Ordiary Dirichlet Serie) Whe,a,,3, are complex umber, we call the followig Ordiary Dirichlet Serie. f() a a a a 3 3 a 4 4 Note
More informationZeta-reciprocal Extended reciprocal zeta function and an alternate formulation of the Riemann hypothesis By M. Aslam Chaudhry
Zeta-reciprocal Eteded reciprocal zeta fuctio ad a alterate formulatio of the Riema hypothei By. Alam Chaudhry Departmet of athematical Sciece, Kig Fahd Uiverity of Petroleum ad ieral Dhahra 36, Saudi
More informationA Faster Product for π and a New Integral for ln π 2
A Fater Product for ad a New Itegral for l Joatha Sodow. INTRODUCTION. I [5] we derived a ifiite product repreetatio of e γ, where γ i Euler cotat: e γ = 3 3 3 4 3 3 Here the th factor i the ( + )th root
More informationFig. 1: Streamline coordinates
1 Equatio of Motio i Streamlie Coordiate Ai A. Soi, MIT 2.25 Advaced Fluid Mechaic Euler equatio expree the relatiohip betwee the velocity ad the preure field i ivicid flow. Writte i term of treamlie coordiate,
More informationSTRONG DEVIATION THEOREMS FOR THE SEQUENCE OF CONTINUOUS RANDOM VARIABLES AND THE APPROACH OF LAPLACE TRANSFORM
Joural of Statitic: Advace i Theory ad Applicatio Volume, Number, 9, Page 35-47 STRONG DEVIATION THEORES FOR THE SEQUENCE OF CONTINUOUS RANDO VARIABLES AND THE APPROACH OF LAPLACE TRANSFOR School of athematic
More informationu t u 0 ( 7) Intuitively, the maximum principles can be explained by the following observation. Recall
Oct. Heat Equatio M aximum priciple I thi lecture we will dicu the maximum priciple ad uiquee of olutio for the heat equatio.. Maximum priciple. The heat equatio alo ejoy maximum priciple a the Laplace
More informationApplied Mathematical Sciences, Vol. 9, 2015, no. 3, HIKARI Ltd,
Applied Mathematical Sciece Vol 9 5 o 3 7 - HIKARI Ltd wwwm-hiaricom http://dxdoiorg/988/am54884 O Poitive Defiite Solutio of the Noliear Matrix Equatio * A A I Saa'a A Zarea* Mathematical Sciece Departmet
More informationFractional parts and their relations to the values of the Riemann zeta function
Arab. J. Math. (08) 7: 8 http://doi.org/0.007/40065-07-084- Arabia Joural of Mathematic Ibrahim M. Alabdulmohi Fractioal part ad their relatio to the value of the Riema zeta fuctio Received: 4 Jauary 07
More informationx z Increasing the size of the sample increases the power (reduces the probability of a Type II error) when the significance level remains fixed.
] z-tet for the mea, μ If the P-value i a mall or maller tha a pecified value, the data are tatitically igificat at igificace level. Sigificace tet for the hypothei H 0: = 0 cocerig the ukow mea of a populatio
More informationComments on Discussion Sheet 18 and Worksheet 18 ( ) An Introduction to Hypothesis Testing
Commet o Dicuio Sheet 18 ad Workheet 18 ( 9.5-9.7) A Itroductio to Hypothei Tetig Dicuio Sheet 18 A Itroductio to Hypothei Tetig We have tudied cofidece iterval for a while ow. Thee are method that allow
More informationNew integral representations. . The polylogarithm function
New itegral repreetatio of the polylogarithm fuctio Djurdje Cvijović Atomic Phyic Laboratory Viča Ititute of Nuclear Sciece P.O. Box 5 Belgrade Serbia. Abtract. Maximo ha recetly give a excellet ummary
More information(a 1 ) n (a p ) n z n (b 1 ) n (b q ) n n!, (1)
MATEMATIQKI VESNIK 64, 3 (01), 40 45 September 01 origiali auqi rad reearch paper INTEGRAL AND COMPUTATIONAL REPRESENTATION OF SUMMATION WHICH EXTENDS A RAMANUJAN S SUM Tibor K. Pogáy, Arju K. Rathie ad
More informationOn the Signed Domination Number of the Cartesian Product of Two Directed Cycles
Ope Joural of Dicrete Mathematic, 205, 5, 54-64 Publihed Olie July 205 i SciRe http://wwwcirporg/oural/odm http://dxdoiorg/0426/odm2055005 O the Siged Domiatio Number of the Carteia Product of Two Directed
More informationTESTS OF SIGNIFICANCE
TESTS OF SIGNIFICANCE Seema Jaggi I.A.S.R.I., Library Aveue, New Delhi eema@iari.re.i I applied ivetigatio, oe i ofte itereted i comparig ome characteritic (uch a the mea, the variace or a meaure of aociatio
More informationSociété de Calcul Mathématique, S. A. Algorithmes et Optimisation
Société de Calcul Mathématique S A Algorithme et Optimiatio Radom amplig of proportio Berard Beauzamy Jue 2008 From time to time we fid a problem i which we do ot deal with value but with proportio For
More informationExpectation of the Ratio of a Sum of Squares to the Square of the Sum : Exact and Asymptotic results
Expectatio of the Ratio of a Sum of Square to the Square of the Sum : Exact ad Aymptotic reult A. Fuch, A. Joffe, ad J. Teugel 63 October 999 Uiverité de Strabourg Uiverité de Motréal Katholieke Uiveriteit
More informationHeat Equation: Maximum Principles
Heat Equatio: Maximum Priciple Nov. 9, 0 I thi lecture we will dicu the maximum priciple ad uiquee of olutio for the heat equatio.. Maximum priciple. The heat equatio alo ejoy maximum priciple a the Laplace
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 16 11/04/2013. Ito integral. Properties
MASSACHUSES INSIUE OF ECHNOLOGY 6.65/15.7J Fall 13 Lecture 16 11/4/13 Ito itegral. Propertie Cotet. 1. Defiitio of Ito itegral. Propertie of Ito itegral 1 Ito itegral. Exitece We cotiue with the cotructio
More informationON THE SCALE PARAMETER OF EXPONENTIAL DISTRIBUTION
Review of the Air Force Academy No. (34)/7 ON THE SCALE PARAMETER OF EXPONENTIAL DISTRIBUTION Aca Ileaa LUPAŞ Military Techical Academy, Bucharet, Romaia (lua_a@yahoo.com) DOI:.96/84-938.7.5..6 Abtract:
More informationA Tail Bound For Sums Of Independent Random Variables And Application To The Pareto Distribution
Applied Mathematic E-Note, 9009, 300-306 c ISSN 1607-510 Available free at mirror ite of http://wwwmaththuedutw/ ame/ A Tail Boud For Sum Of Idepedet Radom Variable Ad Applicatio To The Pareto Ditributio
More informationChapter 9. Key Ideas Hypothesis Test (Two Populations)
Chapter 9 Key Idea Hypothei Tet (Two Populatio) Sectio 9-: Overview I Chapter 8, dicuio cetered aroud hypothei tet for the proportio, mea, ad tadard deviatio/variace of a igle populatio. However, ofte
More informationEULER-MACLAURIN SUM FORMULA AND ITS GENERALIZATIONS AND APPLICATIONS
EULER-MACLAURI SUM FORMULA AD ITS GEERALIZATIOS AD APPLICATIOS Joe Javier Garcia Moreta Graduate tudet of Phyic at the UPV/EHU (Uiverity of Baque coutry) I Solid State Phyic Addre: Practicate Ada y Grijalba
More informationREVIEW OF SIMPLE LINEAR REGRESSION SIMPLE LINEAR REGRESSION
REVIEW OF SIMPLE LINEAR REGRESSION SIMPLE LINEAR REGRESSION I liear regreio, we coider the frequecy ditributio of oe variable (Y) at each of everal level of a ecod variable (X). Y i kow a the depedet variable.
More informationOn Elementary Methods to Evaluate Values of the Riemann Zeta Function and another Closely Related Infinite Series at Natural Numbers
Global oural of Mathematical Sciece: Theory a Practical. SSN 97- Volume 5, Number, pp. 5-59 teratioal Reearch Publicatio Houe http://www.irphoue.com O Elemetary Metho to Evaluate Value of the Riema Zeta
More informationM227 Chapter 9 Section 1 Testing Two Parameters: Means, Variances, Proportions
M7 Chapter 9 Sectio 1 OBJECTIVES Tet two mea with idepedet ample whe populatio variace are kow. Tet two variace with idepedet ample. Tet two mea with idepedet ample whe populatio variace are equal Tet
More informationA tail bound for sums of independent random variables : application to the symmetric Pareto distribution
A tail boud for um of idepedet radom variable : applicatio to the ymmetric Pareto ditributio Chritophe Cheeau To cite thi verio: Chritophe Cheeau. A tail boud for um of idepedet radom variable : applicatio
More informationOn the 2-Domination Number of Complete Grid Graphs
Ope Joural of Dicrete Mathematic, 0,, -0 http://wwwcirporg/oural/odm ISSN Olie: - ISSN Prit: - O the -Domiatio Number of Complete Grid Graph Ramy Shahee, Suhail Mahfud, Khame Almaea Departmet of Mathematic,
More informationDISCRETE MELLIN CONVOLUTION AND ITS EXTENSIONS, PERRON FORMULA AND EXPLICIT FORMULAE
DISCRETE MELLIN CONVOLUTION AND ITS EXTENSIONS, PERRON FORMULA AND EXPLICIT FORMULAE Joe Javier Garcia Moreta Graduate tudet of Phyic at the UPV/EHU (Uiverity of Baque coutry) I Solid State Phyic Addre:
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationSequences of Definite Integrals, Factorials and Double Factorials
47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry Daa-Picard Departmet of Applied Mathematics Jerusalem College of Techology
More informationState space systems analysis
State pace ytem aalyi Repreetatio of a ytem i tate-pace (tate-pace model of a ytem To itroduce the tate pace formalim let u tart with a eample i which the ytem i dicuio i a imple electrical circuit with
More informationMath 213b (Spring 2005) Yum-Tong Siu 1. Explicit Formula for Logarithmic Derivative of Riemann Zeta Function
Math 3b Sprig 005 Yum-og Siu Expliit Formula for Logarithmi Derivative of Riema Zeta Futio he expliit formula for the logarithmi derivative of the Riema zeta futio i the appliatio to it of the Perro formula
More informationGeneralized Likelihood Functions and Random Measures
Pure Mathematical Sciece, Vol. 3, 2014, o. 2, 87-95 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/pm.2014.437 Geeralized Likelihood Fuctio ad Radom Meaure Chrito E. Koutzaki Departmet of Mathematic
More informationSOLUTION: The 95% confidence interval for the population mean µ is x ± t 0.025; 49
C22.0103 Sprig 2011 Homework 7 olutio 1. Baed o a ample of 50 x-value havig mea 35.36 ad tadard deviatio 4.26, fid a 95% cofidece iterval for the populatio mea. SOLUTION: The 95% cofidece iterval for the
More informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
More informationStatistics and Chemical Measurements: Quantifying Uncertainty. Normal or Gaussian Distribution The Bell Curve
Statitic ad Chemical Meauremet: Quatifyig Ucertaity The bottom lie: Do we trut our reult? Should we (or ayoe ele)? Why? What i Quality Aurace? What i Quality Cotrol? Normal or Gauia Ditributio The Bell
More informationLECTURE 13 SIMULTANEOUS EQUATIONS
NOVEMBER 5, 26 Demad-upply ytem LETURE 3 SIMULTNEOUS EQUTIONS I thi lecture, we dicu edogeeity problem that arie due to imultaeity, i.e. the left-had ide variable ad ome of the right-had ide variable are
More informationStatistical Inference Procedures
Statitical Iferece Procedure Cofidece Iterval Hypothei Tet Statitical iferece produce awer to pecific quetio about the populatio of iteret baed o the iformatio i a ample. Iferece procedure mut iclude a
More informationAppendix to Quicksort Asymptotics
Appedix to Quicksort Asymptotics James Alle Fill Departmet of Mathematical Scieces The Johs Hopkis Uiversity jimfill@jhu.edu ad http://www.mts.jhu.edu/~fill/ ad Svate Jaso Departmet of Mathematics Uppsala
More informationWe will look for series solutions to (1) around (at most) regular singular points, which without
ENM 511 J. L. Baai April, 1 Frobeiu Solutio to a d order ODE ear a regular igular poit Coider the ODE y 16 + P16 y 16 + Q1616 y (1) We will look for erie olutio to (1) aroud (at mot) regular igular poit,
More informationOn The Computation Of Weighted Shapley Values For Cooperative TU Games
O he Computatio Of Weighted hapley Value For Cooperative U Game Iriel Draga echical Report 009-0 http://www.uta.edu/math/preprit/ Computatio of Weighted hapley Value O HE COMPUAIO OF WEIGHED HAPLEY VALUE
More informationarxiv: v1 [math.pr] 31 Jan 2012
EXACT L 2 -DISTANCE FROM THE LIMIT FOR QUICKSORT KEY COMPARISONS EXTENDED ABSTRACT) arxiv:20.6445v [math.pr] 3 Ja 202 PATRICK BINDJEME JAMES ALLEN FILL Abstract Usigarecursiveapproach, weobtaiasimpleexactexpressioforthel
More informationGeneralized Fibonacci Like Sequence Associated with Fibonacci and Lucas Sequences
Turkih Joural of Aalyi ad Number Theory, 4, Vol., No. 6, 33-38 Available olie at http://pub.ciepub.com/tjat//6/9 Sciece ad Educatio Publihig DOI:.69/tjat--6-9 Geeralized Fiboacci Like Sequece Aociated
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 30 Sigal & Sytem Prof. Mark Fowler Note Set #8 C-T Sytem: Laplace Traform Solvig Differetial Equatio Readig Aigmet: Sectio 6.4 of Kame ad Heck / Coure Flow Diagram The arrow here how coceptual flow
More informationWeak formulation and Lagrange equations of motion
Chapter 4 Weak formulatio ad Lagrage equatio of motio A mot commo approach to tudy tructural dyamic i the ue of the Lagrage equatio of motio. Thee are obtaied i thi chapter tartig from the Cauchy equatio
More informationThe Gamma function Michael Taylor. Abstract. This material is excerpted from 18 and Appendix J of [T].
The Gamma fuctio Michael Taylor Abstract. This material is excerpted from 8 ad Appedix J of [T]. The Gamma fuctio has bee previewed i 5.7 5.8, arisig i the computatio of a atural Laplace trasform: 8. ft
More information3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,
3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [
More informationSTA 4032 Final Exam Formula Sheet
Chapter 2. Probability STA 4032 Fial Eam Formula Sheet Some Baic Probability Formula: (1) P (A B) = P (A) + P (B) P (A B). (2) P (A ) = 1 P (A) ( A i the complemet of A). (3) If S i a fiite ample pace
More informationSequences. Notation. Convergence of a Sequence
Sequeces A sequece is essetially just a list. Defiitio (Sequece of Real Numbers). A sequece of real umbers is a fuctio Z (, ) R for some real umber. Do t let the descriptio of the domai cofuse you; it
More informationELEC 372 LECTURE NOTES, WEEK 4 Dr. Amir G. Aghdam Concordia University
ELEC 37 LECTURE NOTES, WEE 4 Dr Amir G Aghdam Cocordia Uiverity Part of thee ote are adapted from the material i the followig referece: Moder Cotrol Sytem by Richard C Dorf ad Robert H Bihop, Pretice Hall
More informationLast time: Completed solution to the optimum linear filter in real-time operation
6.3 tochatic Etimatio ad Cotrol, Fall 4 ecture at time: Completed olutio to the oimum liear filter i real-time operatio emi-free cofiguratio: t D( p) F( p) i( p) dte dp e π F( ) F( ) ( ) F( p) ( p) 4444443
More informationLecture 30: Frequency Response of Second-Order Systems
Lecture 3: Frequecy Repoe of Secod-Order Sytem UHTXHQF\ 5HVSRQVH RI 6HFRQGUGHU 6\VWHPV A geeral ecod-order ytem ha a trafer fuctio of the form b + b + b H (. (9.4 a + a + a It ca be table, utable, caual
More informationPerformance-Based Plastic Design (PBPD) Procedure
Performace-Baed Platic Deig (PBPD) Procedure 3. Geeral A outlie of the tep-by-tep, Performace-Baed Platic Deig (PBPD) procedure follow, with detail to be dicued i ubequet ectio i thi chapter ad theoretical
More informationTHE CONCEPT OF THE ROOT LOCUS. H(s) THE CONCEPT OF THE ROOT LOCUS
So far i the tudie of cotrol yte the role of the characteritic equatio polyoial i deteriig the behavior of the yte ha bee highlighted. The root of that polyoial are the pole of the cotrol yte, ad their
More informationExplicit scheme. Fully implicit scheme Notes. Fully implicit scheme Notes. Fully implicit scheme Notes. Notes
Explicit cheme So far coidered a fully explicit cheme to umerically olve the diffuio equatio: T + = ( )T + (T+ + T ) () with = κ ( x) Oly table for < / Thi cheme i ometime referred to a FTCS (forward time
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationOn Random Line Segments in the Unit Square
O Radom Lie Segmets i the Uit Square Thomas A. Courtade Departmet of Electrical Egieerig Uiversity of Califoria Los Ageles, Califoria 90095 Email: tacourta@ee.ucla.edu I. INTRODUCTION Let Q = [0, 1] [0,
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationChapter 8. Euler s Gamma function
Chapter 8 Euler s Gamma fuctio The Gamma fuctio plays a importat role i the fuctioal equatio for ζ(s that we will derive i the ext chapter. I the preset chapter we have collected some properties of the
More informationSTUDENT S t-distribution AND CONFIDENCE INTERVALS OF THE MEAN ( )
STUDENT S t-distribution AND CONFIDENCE INTERVALS OF THE MEAN Suppoe that we have a ample of meaured value x1, x, x3,, x of a igle uow quatity. Aumig that the meauremet are draw from a ormal ditributio
More informationNew proofs of the duplication and multiplication formulae for the gamma and the Barnes double gamma functions. Donal F. Connon
New proof of the duplicatio ad multiplicatio formulae for the gamma ad the Bare double gamma fuctio Abtract Doal F. Coo dcoo@btopeworld.com 6 March 9 New proof of the duplicatio formulae for the gamma
More informationBrief Review of Linear System Theory
Brief Review of Liear Sytem heory he followig iformatio i typically covered i a coure o liear ytem theory. At ISU, EE 577 i oe uch coure ad i highly recommeded for power ytem egieerig tudet. We have developed
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationConvergence of random variables. (telegram style notes) P.J.C. Spreij
Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space
More information1 Approximating Integrals using Taylor Polynomials
Seughee Ye Ma 8: Week 7 Nov Week 7 Summary This week, we will lear how we ca approximate itegrals usig Taylor series ad umerical methods. Topics Page Approximatig Itegrals usig Taylor Polyomials. Defiitios................................................
More informationMAT 271 Project: Partial Fractions for certain rational functions
MAT 7 Project: Partial Fractios for certai ratioal fuctios Prerequisite kowledge: partial fractios from MAT 7, a very good commad of factorig ad complex umbers from Precalculus. To complete this project,
More informationChapter 8. Euler s Gamma function
Chapter 8 Euler s Gamma fuctio The Gamma fuctio plays a importat role i the fuctioal equatio for ζ(s) that we will derive i the ext chapter. I the preset chapter we have collected some properties of the
More informationChapter 10: Power Series
Chapter : Power Series 57 Chapter Overview: Power Series The reaso series are part of a Calculus course is that there are fuctios which caot be itegrated. All power series, though, ca be itegrated because
More informationThe natural exponential function
The atural expoetial fuctio Attila Máté Brookly College of the City Uiversity of New York December, 205 Cotets The atural expoetial fuctio for real x. Beroulli s iequality.....................................2
More informationComplex Analysis Spring 2001 Homework I Solution
Complex Aalysis Sprig 2001 Homework I Solutio 1. Coway, Chapter 1, sectio 3, problem 3. Describe the set of poits satisfyig the equatio z a z + a = 2c, where c > 0 ad a R. To begi, we see from the triagle
More informationRiemann Paper (1859) Is False
Riema Paper (859) I Fale Chu-Xua Jiag P O Box94, Beijig 00854, Chia Jiagchuxua@vipohucom Abtract I 859 Riema defied the zeta fuctio ζ () From Gamma fuctio he derived the zeta fuctio with Gamma fuctio ζ
More informationCOMPARISONS INVOLVING TWO SAMPLE MEANS. Two-tail tests have these types of hypotheses: H A : 1 2
Tetig Hypothee COMPARISONS INVOLVING TWO SAMPLE MEANS Two type of hypothee:. H o : Null Hypothei - hypothei of o differece. or 0. H A : Alterate Hypothei hypothei of differece. or 0 Two-tail v. Oe-tail
More informationCapacitors and PN Junctions. Lecture 8: Prof. Niknejad. Department of EECS University of California, Berkeley. EECS 105 Fall 2003, Lecture 8
CS 15 Fall 23, Lecture 8 Lecture 8: Capacitor ad PN Juctio Prof. Nikejad Lecture Outlie Review of lectrotatic IC MIM Capacitor No-Liear Capacitor PN Juctio Thermal quilibrium lectrotatic Review 1 lectric
More informationLinear Regression Demystified
Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to
More informationIntroduction to Control Systems
Itroductio to Cotrol Sytem CLASSIFICATION OF MATHEMATICAL MODELS Icreaig Eae of Aalyi Static Icreaig Realim Dyamic Determiitic Stochatic Lumped Parameter Ditributed Parameter Liear Noliear Cotat Coefficiet
More informationThe Riemann Zeta Function
Physics 6A Witer 6 The Riema Zeta Fuctio I this ote, I will sketch some of the mai properties of the Riema zeta fuctio, ζ(x). For x >, we defie ζ(x) =, x >. () x = For x, this sum diverges. However, we
More informationChapter 1. Complex Numbers. Dr. Pulak Sahoo
Chapter 1 Complex Numbers BY Dr. Pulak Sahoo Assistat Professor Departmet of Mathematics Uiversity Of Kalyai West Begal, Idia E-mail : sahoopulak1@gmail.com 1 Module-2: Stereographic Projectio 1 Euler
More informationMath 525: Lecture 5. January 18, 2018
Math 525: Lecture 5 Jauary 18, 2018 1 Series (review) Defiitio 1.1. A sequece (a ) R coverges to a poit L R (writte a L or lim a = L) if for each ǫ > 0, we ca fid N such that a L < ǫ for all N. If the
More informationThe log-behavior of n p(n) and n p(n)/n
Ramauja J. 44 017, 81-99 The log-behavior of p ad p/ William Y.C. Che 1 ad Ke Y. Zheg 1 Ceter for Applied Mathematics Tiaji Uiversity Tiaji 0007, P. R. Chia Ceter for Combiatorics, LPMC Nakai Uivercity
More informationIntroEcono. Discrete RV. Continuous RV s
ItroEcoo Aoc. Prof. Poga Porchaiwiekul, Ph.D... ก ก e-mail: Poga.P@chula.ac.th Homepage: http://pioeer.chula.ac.th/~ppoga (c) Poga Porchaiwiekul, Chulalogkor Uiverity Quatitative, e.g., icome, raifall
More informationENGI Series Page 6-01
ENGI 3425 6 Series Page 6-01 6. Series Cotets: 6.01 Sequeces; geeral term, limits, covergece 6.02 Series; summatio otatio, covergece, divergece test 6.03 Stadard Series; telescopig series, geometric series,
More informationChapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
More information100(1 α)% confidence interval: ( x z ( sample size needed to construct a 100(1 α)% confidence interval with a margin of error of w:
Stat 400, ectio 7. Large Sample Cofidece Iterval ote by Tim Pilachowki a Large-Sample Two-ided Cofidece Iterval for a Populatio Mea ectio 7.1 redux The poit etimate for a populatio mea µ will be a ample
More information10-716: Advanced Machine Learning Spring Lecture 13: March 5
10-716: Advaced Machie Learig Sprig 019 Lecture 13: March 5 Lecturer: Pradeep Ravikumar Scribe: Charvi Ratogi, Hele Zhou, Nicholay opi Note: Lae template courtey of UC Berkeley EECS dept. Diclaimer: hee
More informationHarmonic Number Identities Via Euler s Transform
1 2 3 47 6 23 11 Joural of Iteger Sequeces, Vol. 12 2009), Article 09.6.1 Harmoic Number Idetities Via Euler s Trasform Khristo N. Boyadzhiev Departmet of Mathematics Ohio Norther Uiversity Ada, Ohio 45810
More information1. (a) If u (I : R J), there exists c 0 in R such that for all q 0, cu q (I : R J) [q] I [q] : R J [q]. Hence, if j J, for all q 0, j q (cu q ) =
Math 615, Witer 2016 Problem Set #5 Solutio 1. (a) If u (I : R J), there exit c 0 i R uch that for all q 0, cu q (I : R J) [q] I [q] : R J [q]. Hece, if j J, for all q 0, j q (cu q ) = c(ju) q I [q], o
More informationC. Complex Numbers. x 6x + 2 = 0. This equation was known to have three real roots, given by simple combinations of the expressions
C. Complex Numbers. Complex arithmetic. Most people thik that complex umbers arose from attempts to solve quadratic equatios, but actually it was i coectio with cubic equatios they first appeared. Everyoe
More informationCHAPTER 5. Theory and Solution Using Matrix Techniques
A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL
More informationINFINITE SEQUENCES AND SERIES
11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES 11.4 The Compariso Tests I this sectio, we will lear: How to fid the value of a series by comparig it with a kow series. COMPARISON TESTS
More informationIsolated Word Recogniser
Lecture 5 Iolated Word Recogitio Hidde Markov Model of peech State traitio ad aligmet probabilitie Searchig all poible aligmet Dyamic Programmig Viterbi Aligmet Iolated Word Recogitio 8. Iolated Word Recogier
More informationCOMPUTING THE EULER S CONSTANT: A HISTORICAL OVERVIEW OF ALGORITHMS AND RESULTS
COMPUTING THE EULER S CONSTANT: A HISTORICAL OVERVIEW OF ALGORITHMS AND RESULTS GONÇALO MORAIS Abstract. We preted to give a broad overview of the algorithms used to compute the Euler s costat. Four type
More informationTHE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS
THE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS DEMETRES CHRISTOFIDES Abstract. Cosider a ivertible matrix over some field. The Gauss-Jorda elimiatio reduces this matrix to the idetity
More informationBernoulli numbers and the Euler-Maclaurin summation formula
Physics 6A Witer 006 Beroulli umbers ad the Euler-Maclauri summatio formula I this ote, I shall motivate the origi of the Euler-Maclauri summatio formula. I will also explai why the coefficiets o the right
More informationCollective Support Recovery for Multi-Design Multi-Response Linear Regression
IEEE TRANACTION ON INFORMATION THEORY, VOL XX, NO XX, XX 4 Collective upport Recovery for Multi-Deig Multi-Repoe Liear Regreio eiguag ag, Yigbi Liag, Eric P Xig Abtract The multi-deig multi-repoe MDMR
More informationDirichlet s Theorem on Arithmetic Progressions
Dirichlet s Theorem o Arithmetic Progressios Athoy Várilly Harvard Uiversity, Cambridge, MA 0238 Itroductio Dirichlet s theorem o arithmetic progressios is a gem of umber theory. A great part of its beauty
More informationAssignment 1 - Solutions. ECSE 420 Parallel Computing Fall November 2, 2014
Aigmet - Solutio ECSE 420 Parallel Computig Fall 204 ovember 2, 204. (%) Decribe briefly the followig term, expoe their caue, ad work-aroud the idutry ha udertake to overcome their coequece: (i) Memory
More informationA Note on the Symmetric Powers of the Standard Representation of S n
A Note o the Symmetric Powers of the Stadard Represetatio of S David Savitt 1 Departmet of Mathematics, Harvard Uiversity Cambridge, MA 0138, USA dsavitt@mathharvardedu Richard P Staley Departmet of Mathematics,
More information