Combinatorial Interpretation of Raney Numbers and Tree Enumerations

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Combinatorial Interpretation of Raney Numbers and Tree Enumerations"

Transcription

1 Ope Joual of Discete Matheatics, 2015, 5, 1-9 Published Olie Jauay 2015 i SciRes. Cobiatoial Itepetatio of Raey Nubes ad Tee Eueatios Chi Hee Pah 1, Mohaed Ridza Wahiddi 2 1 Depatet of Coputatioal ad Theoetical Scieces, Faculty of Sciece, Iteatioal Islaic Uivesity Malaysia, Kuata, Malaysia 2 Depatet of Copute Sciece, Faculty of ICT, Iteatioal Islaic Uivesity Malaysia, Kuala Lupu, Malaysia Eail: Received 24 Novebe 2014; evised 20 Decebe 2014; accepted 3 Jauay 2015 Copyight 2015 by authos ad Scietific Reseach Publishig Ic. This wo is licesed ude the Ceative Coos Attibutio Iteatioal Licese (CC BY). Abstact A ew cobiatoial itepetatio of Raey ubes is poposed. We apply this cobiatoial itepetatio to solve seveal tee eueatio coutig pobles. Futhe a geealized Catala tiagle is itoduced ad soe of its popeties ae poved. Keywods Raey Nubes, Fuss-Catala Nubes, Tee Eueatio, Netwo 1. Itoductio Iteestigly Peso ad Zyczowsi wee the fist who use the te Raey ubes [1] [2] ad it is defied as + R (, ) = whee 2, 1, 1. Nevetheless, it is ow that Raey s lea could be + used i coutig poble associated with Catala ubes [3] ad a bijectio exists betwee Raey path ad pla ultitee [4]. These ubes do ot fo ovel sequeces, as the ubes wee itoduced ealie as a geealizatio of the bioial seies [5]. Moeove, the sequece R4 (, 5) = 1, 5, 30, 200,1425,10626,81900, , is ot icluded i OEIS database [6] befoe If we let = 1, we obtai aothe ow sequece, i.e., 1 Fuss-Catala ubes [7] [8] which is defied as C ( ) =. Although Fuss-Catala ubes ( 1) + 1 How to cite this pape: Pah, C.H. ad Wahiddi, M.R. (2015) Cobiatoial Itepetatio of Raey Nubes ad Tee Eueatios. Ope Joual of Discete Matheatics, 5,

2 C. H. Pah, M. R. Wahiddi wee itoduced ealie tha Catala ubes [9], the Catala ubes ae oe popula ad widely used tha the Fuss-Catala ubes (see [10] [11] fo details). Due to its self siila stuctue, the applicatios of Catala ubes could be foud i ay physical pobles, e.g., lattice odel [12], tee eueatio etwo [13], ad Hael atices i codig theoy [14]. A tee is a coected gaph with o cycles ad fo which oly oe shotest path exists fo oe ode to aothe. Tee eueatio is a ipotat tool to study etwo. These etwos always gow i a powe-law behavio which is ofte foud i social etwo, subway syste [15], etc. R i the fo of a o-liea ecusio ad the we povide a cobiatoial itepetatio of Raey ubes. Usig this cobiatoial itepetatio, we solve seveal tee eueatio coutig pobles i which we ecove the well-ow Fuss-Catala ubes [16], Catala tiagles [17], ad othe less ow ubes. Motivated by the coectio betwee Raey ubes ad Catala tiagles, a geealizatio of Catala tiagles is poposed ad we pove soe of thei popeties. Cosequetly these foulas geealize the popeties of Catala tiagles. Fo the exact solutio of these tee eueatio pobles, we ae able to fid a shap uppe boud of the ube of each tee eueatio poble. The uppe boud is ipotat i the cotou ethod fo lattice odels ad liit of the ado gaph. I this pape, we itoduce Raey ubes (, ) 2. Raey Nubes Let C ( ) be the ube of a -ay tees with labeled vetices (Figue 1), whee 1 C ( ) =, 2, 1. ( 1) + 1 The Raey ubes ae defied as follows: R, = C i C i C i, C 0 = 1, > 0 (1) whee i, i,, i { 0} ( ) ( 1) ( 2) ( ) ( ) i1+ i2+ + i = 1 2. Theefoe, the cobiatoial itepetatio is as follows: copies of -ay tee with total ube of vetices. Next, we let u ( x ) be the geeatig fuctio fo C ( ), i.e., 2 u x = 1+ C 1 x+ C 2 x + + C x +. ( ) ( ) ( ) ( ) The, the geeatig fuctio of R (, ) is u ( ) [18]. Lea 1. Let u ( ) x be the geeatig fuctio of the Raey ubes. The, ( ( )) + x u x =. + x ad the Raey ubes satisfy the followig foula Iediately, we obtai the followig theoe. Theoe 1. The bioial fos of the Raey ubes ae give by + R (, ) =. + Fo theoe (1), it is ot difficult to deduce soe of the popeties of Raey ubes. Coollay 1. Fo itege > 1, we have ( ) R 0, = 1; (4) (2) (3) Figue 1. A biay tee with 3 odes, whee the botto vetex is the oot. 2

3 C. H. Pah, M. R. Wahiddi i R ( i, i) = ; i 1 R ( 1, ) = R (,1 ) = ; ( 1) + 1 R + 1, =. 1 ( ) Coollay 2. We ca wite C ( ) i a oliea ecusio as: C ( ) C ( i1) C ( i2) C ( i) C ( ) whee i, i,, i { 0} + 1 =, 0 = 1 (8) i1+ i2+ + i = 1 2. We ecove the foula by joiig the copies of -ay tee with vetices which is also equivalet to a -ay tee with vetices ad a additioal oot (see Figue 2): Usig bioial fo of R (, ) ( ) = ( + ) R, R 1,1., oe ca obtai the followig esult. Coollay 3. Fo a fixed itege > 1, ad > 1, whee ( ) ( 1, 1 ) (, ) (, 1) R + = R R (9) R, = 0 if <. Fo = 2, we ecove the idetity of a geealized Ballot ubes: 3. A Hoogeeous -Ay Tee ( ) ( ) ( ) R 1, + 1 = R, R, 1. (10) Ulie the usual -ay tee, we defie a hoogeeous -ay tee as a gaph with o cycles, i which each vetex eaates + 1 edges (see Figue 3 fo = 4). We fix a vetex aely z as the oot. Ulie the odiay oot i a -ay tee, this oot has + 1 successos while othe vetices have ube of successos. Ay vetex could be chose to be the oot sice the gaph is hoogeous. Fo a give vetices, we ay fid how ay coected sub-tee ooted at z. This ube is defied as D ( ). Theoe 2. Fo 1 C as: >, we ca wite D ( ) i a oliea ecusio of ( ) ( ) ( ) ( ) D = C C fo > 0. (11) = 1 Poof. We decopose the poble by fidig out the ube of -ay tee of ube of oe copy of C, ad aothe copy of -ay tee with C. -ay tee with vetices, i.e., ( ) (5) (6) (7) vetices, i.e. ( ) Figue 2. Joiig 4 ooted Cayley tee of ode 4 whee = 4 ad = 4. 3

4 C. H. Pah, M. R. Wahiddi Figue 3. A hoogeous gaph whee each vetex is coected to exactly 5 eighbous. x, its age should be fo 1 to. Total D ( ) 0 Sice the foe C ust always iclude of all C ( ) C ( ) usig the additio ad ultiplicatio piciples. Usig Equatio (8), we ewite the foula above as ( ) ( 1) ( 2) ( + 1) = 1 is just the su D = C C C fo > 0. (12) This foula ca also be obtaied usig + 1 copies of -ay tee togethe with 1 vetices ad oe cete. We the fid the bioial fo of D ( ). Coollay 4. Fo 2, D ( ) is expessed i bioial fo as: + 1 D ( ) = R ( 1, + 1) = > 0. (13) ( 1) Thus, D ( ) = ad D ( ) = as i [19] ad [20], espectively. Fo = 4, D ( ) coicides with oe fo of Raey ubes as etioed above, i.e., R4 (,5). The ubes D ( ) a lot of ew sequeces. Fo exaple, the sequece of R ( ), i.e., is ot foud i the OEIS database [6]. Fo theoe (2), oe ca get 5,6 1,6, 45,380,3450,32886,324632, , , , ( ) = ( ) ( ) ( ) D C C C = 0. 4 geeate This foula ca be also obtaied easily by a diffeet way: 1) Cout the ube of tees by joiig the 2 copies of ay tee, with total ube of vetices. 2) Subtact those tees eueate fo y but does t cotai z, that is, exactly the ube C ( ). Let the geeatig fuctio of D ( ) be w ( x ). The we have the followig esult. Coollay 5. Fo > 1 ad 0 w x is whee ( ), the geeatig fuctio, ( ) 2 w ( x) u ( x) u ( x) u x is the geeatig fuctio of C ( ). Coollay 6. Fo > 1, = (14) 4

5 C. H. Pah, M. R. Wahiddi o ( ) ( ) = ( 1, + 1) C C R (15) = 1 = 1 ( ) ( ) = ( + ) R,1 R,1 R 1, 1. (16) Usig the bioial iequality i [21] ad the bioial fos of ( ) equality ca be easily poved. Coollay 7. Fo 2 >, whee b = 1 1 ad > 0. ( ) ( ) ( b ) 32 C ad D ( ), the followig i- C D < (17) Fo sufficietly lage, a siple fo is poduced as well, i.e., C ( ) D ( ) cojectued i a weae fo i [20], i.e., C ( ) D ( ) 4. Catala Tiagle A Catala tiagle B(, ) is defied as follows [9]: The Catala tiagle satisfies [9]: ( e) <. ( e) 1 if = = 1; B(, ) = B( 1, 1) + 2B( 1, ) + B( 1, + 1) if 1 ; 0 othewise. ( ) 2( 1) 2( 2) 2( ) 2( ) i1 + + i = i1,, i 1 <. These esults ae 32 B, = C i C i C i, C 0 = 1 > 0 (18) Usig a popety of Catala ubes, C ( i ) C ( j ) C ( j ) fo of B(, ), i.e., whee j1, j2,, j2 { 0} ubes, i.e., B(, ) R(,2) =, whee j 1, j 2 0, we get aothe j1+ j2= i 1 ( ) ( ) ( ) ( ) B, = C j C j C j, (19) j1+ j2+ + j =. Fo Equatio (1), we iediately ecove the Catala tiagle fo the Raey = : 2 2 B(, ) = R2 (,2 ) =. We ow coside the followig poble as i [22]: Fid out the ube of all diffeet coected sub-tees of a hoogeous biay tee with ube of vetices, cotaiig the give ube of fixed vetices (whee 2 2). The coditio, 2 2, is siply the ube of vetices that coves the iial copoet cotaiig all vetices. The details of this poble ad teiologies could be foud i the oigial pape [22]. We deote the solutio to this poble as F. I this pape, we show that a solutio to the case whe the iial copoet is full, is as below: ( + ) 2 2 F = B( + 2, ) =, (20) 5

6 C. H. Pah, M. R. Wahiddi whee is the ube of give vetices ad is the ube of fixed vetices i each of the coected subtee. Now, we itepet ad elate the poble above with the cobiatoial itepetatio of the Raey ubes though the followig steps (see Figue 4): 1) Give vetices; 2) Fill up all the iteio poits, i.e., 2 ; 3) Fill up all the bouday poits, i.e., ; 4) The oly 2+ 2 vetices ae left; 5) Sice each bouday poit has 2 eighbous which is ot a iteio poit, we have 2 boxes; 6) If 2+ 2 vetices ae give, the thee ae 2 boxes of biay tee to be filled. As a esult, the solutio is ( + ) + ( + ) R2 ( 2+ 2, 2 ) = =. 2( 2+ 2) Futheoe, it is atual to defie a geealized Catala tiagle, i.e., -th Catala tiagle usig Fuss- Catala ubes istead of Catala ubes as i Equatio (19): whee ( ) ( ) ( 1) ( 2) ( ) ( ) B, = C i C i C i, C 0 = 1 > 0, (22) i1 + + i = i1,, i 1 B, = 0 if <. Fo the popety of Fuss-Catala ubes, i.e., coollay (2) C i = C j1 C j2 C j whee j1, j2,, j 0 ( ) ( ) ( ) ( ) j1+ j2+ + j = i 1, we fid aothe fo of B (, ), B ( ) C ( j ) C ( j ) C ( j ) whee j, j,, j { 0} j1+ j2+ + j =, (21), =, (23). Agai, fo Equatio (1), we iediately have B (, ) = R (, ) =. Lea 2. Soe popeties of -th Catala tiagles ae as follows: ( ) ( ) (24) B,1 = C, (25) ( ) B, = 1, (26) ( ) ( ) B, 1 = 1. (27) Figue 4. 4 bouday poits (solid cicles) coected to full iial copoet. 6

7 , oe ca show that: Lea 3. Fo > 1 ad 2 Usig the bioial fo of B (, ) If = 2, we ecove B ( 1, 1) + 2B ( 1, ) + B ( 1, + 1 ) = B2( 1, 1) + 2B2( 1, ) + B2( 1, + 1 ) = = B2(, ). C. H. Pah, M. R. Wahiddi Based o the iitial esult, lea (3), we pove the followig assetio by atheatical iductio with espect to. Theoe 3. Fo fixed, whee > 1, ad > 1, whee ( ) (28) + + B ( 1, 1 + i) =, 0 i + (29) B, = 0 if <. Poof. Assetio is tue fo = 2. Assue that it is tue fo, we coside the followig suatios: B ( 1, 1 + i) + B ( 1, + i) 0 i 0 i + + ( + 1) + + = B B i B i B 1 i 0 i + ( + 1) =. + ( + 1) ( 1, 1) + ( 1, 1 + ) + ( 1, + ) + ( 1, + ) B B i B 1 i i 1 + ( + 1) =. + ( + 1) ( 1, 1) + + ( 1, 1 + ) + ( 1, + ) + 1 B B i B 1 i + ( + 1) =. + ( + 1) ( 1, 1) + ( 1, 1 + ) + ( 1, + ) ( 1) ( 1) B ( 1, 1 + i) =. i + + Hece, the assetio is tue fo ay > 1. Coollay 8. Fo fixed, whee > 1, ad > 1, we have + i + + =. 0 i i + (30) Fo =, we have the followig siple esult: Coollay 9. Fo fixed > 1, ad > 1, B (, ) = B ( 1, 1 + i), 0 i (31) 7

8 C. H. Pah, M. R. Wahiddi B, = 0 if <. whee ( ) 5. Bioial Tasfoatio of -th Catala Tiagle Fo 2 ad, we defie a ew ube H ( ) ( ), as: ( ), ( ) ( 1) ( ) ( 1) ( ) H = C j C j C l C l. (32) j1+ + j+ l1+ + l= j1,, j 0, 1,, 1 If j1 = j2 = = j = 0, all the Fuss-Catala ubes should stat at 1, the we ecove the peviously defied -th Catala tiagle. Fo the popety of Fuss-Catala ubes, C ( i) = C ( j1) C ( j2) C ( j), whee j, j, 0, we foud aothe fo of 1 2 Fo Equatio (1), we iediately have j1+ j2+ + j = i 1 ( ), ( ) = ( 1) ( ) ( 1) ( ) H C j C j C l C l. (33) j1+ + j + l1+ + l = j1,, j, l1,, l 0 ( ) ( 0) + + H, ( ) = H, + ( ) = R(, + ) = + ad fo = 0, we ecove the sae foula fo -th Catala tiagle, ( ) ( ) ( ) H, = R (, ) =. Fo theoe (3), H, is obtaied as a esult of bioial tasfoatio of -Catala tiagles. Coollay 10. Fo fixed > 1, whee > 1, ad > 1, B, = 0 if <. whee ( ) 6. Coclusio B i H (34) ( (, ) ) + =, ( ), 0 i (35) I this pape, we have itoduced the cobiatoial itepetatio of Raey ubes to solve vaious tee eueatio coutig pobles. The uppe boud of ay + 1 ode tee eueatio is geeally foud to be ( e) (. We have also show how a ew ube H ) 32, ( ) ay be deived fo the bioial tasfoatio of -th Catala tiagles. Acowledgeets This eseach is fuded by the MOHE gat FRGS The authos ae gateful to aoyous efeee s suggestio ad ipoveet of the pesetatio of this pape. Refeeces [1] Peso, K.A. ad Zyczowsi, K. (2011) Poduct of Giibe Matices: Fuss-Catala ad Raey Distibutios. Physical Review E, 83, Aticle ID: [2] Mlotowsi, W., Peso, K.A. ad Zyczowsi, K. (2013) Desities of the Raey Distibutios. Docueta Matheatica, 18, [3] Jeuisse, R.H. (2008) Raey ad Catala. Discete Matheatics, 308, [4] Dzieiaczu, M. (2014) Eueatios of Plae Tees with Multiple Edges ad Raey Lattice Paths. Discete Mathe- 8

9 C. H. Pah, M. R. Wahiddi atics, 337, [5] Gould, H.W. (1972) Cobiatoial Idetities: A Stadadized Set of Tables Listig 500 Bioial Coefficiet Suatios. Mogatow. [6] (2011) The O-Lie Ecyclopedia of Itege Sequeces. Sequece A [7] Gaha, R.L., Kuth, D.E. ad Patashi, O. (1994) Cocete Matheatics. Addiso-Wesley, Bosto. [8] Hilto, P. ad Pedese, J. (1991) Catala Nubes, Thei Geealizatio, ad Thei Uses. Matheatical Itelligece, 13, [9] Koshy, T. (2008) Catala Nubes with Applicatios. Oxfod Uivesity Pess, USA. [10] Staley, R.P. (2013) Catala Addedu to Eueative Cobiatoics. Vol. 2. [11] Staley, R.P. (1999) Eueative Cobiatoics. Vol. 2, Cabidge Uivesity Pess, Cabidge. [12] Baxte, R.J. (1982) Exactly Solved Models i Statistical Mechaics. Acadeic Pess, Lodo. [13] Goltsev, A.V., Doogovtsev, S.N. ad Medes, J.F.F. (2008) Citical Pheoea i Coplex Netwos. Reviews of Mode Physics, 80, [14] Ta, U. (2001) Soe Aspects of Hael Matices i Codig Theoy ad Cobiatoics. The Electoic Joual of Cobiatoics, 8, [15] Lee, K., Jug, W.S., Pa, J.S. ad Choi, M.Y. (2000) Statistical Aalysis of the Metopolita Seoul Subway Syste: Netwo Stuctue ad Passege Flows. Physica A, 387, [16] Aval, J.C. (2008) Multivaiate Fuss-Catala Nubes. Discete Matheatics, 308, [17] Shapio, L.W. (1976) A Catala Tiagle. Discete Matheatics, 14, [18] Melii, D., Spugoli, R. ad Vei, M.C. (2006) Lagage Ivesio: Whe ad How. Acta Applicadae Matheatica, 94, [19] Pah, C.H. (2008) A Applicatio of Catala Nube o Cayley Tee of Ode 2: Sigle Polygo Coutig. Bulleti of the Malaysia Matheatical Scieces Society, 31, [20] Pah, C.H. (2010) Sigle Polygo Coutig o Cayley Tee of Ode 3. Joual of Statistical Physics, 140, [21] Sasvai, Z. (1999) Iequalities fo Bioial Coefficiets. Joual of Matheatical Aalysis ad Applicatios, 236, [22] Muhoedov, F., Pah, C.H. ad Sabuov, M. (2010) Sigle Polygo Coutig fo Fixed Nodes i Cayley Tee: Two Exteal Cases. Pepit. 9

10

Generalized Fibonacci-Lucas Sequence

Generalized Fibonacci-Lucas Sequence Tuish Joual of Aalysis ad Numbe Theoy, 4, Vol, No 6, -7 Available olie at http://pubssciepubcom/tjat//6/ Sciece ad Educatio Publishig DOI:6/tjat--6- Geealized Fiboacci-Lucas Sequece Bijeda Sigh, Ompaash

More information

SOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES

SOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES #A17 INTEGERS 9 2009), 181-190 SOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES Deick M. Keiste Depatmet of Mathematics, Pe State Uivesity, Uivesity Pak, PA 16802 dmk5075@psu.edu James A. Selles Depatmet

More information

A New Result On A,p n,δ k -Summabilty

A New Result On A,p n,δ k -Summabilty OSR Joual of Matheatics (OSR-JM) e-ssn: 2278-5728, p-ssn:239-765x. Volue 0, ssue Ve. V. (Feb. 204), PP 56-62 www.iosjouals.og A New Result O A,p,δ -Suabilty Ripeda Kua &Aditya Kua Raghuashi Depatet of

More information

A note on random minimum length spanning trees

A note on random minimum length spanning trees A ote o adom miimum legth spaig tees Ala Fieze Miklós Ruszikó Lubos Thoma Depatmet of Mathematical Scieces Caegie Mello Uivesity Pittsbugh PA15213, USA ala@adom.math.cmu.edu, usziko@luta.sztaki.hu, thoma@qwes.math.cmu.edu

More information

EVALUATION OF SUMS INVOLVING GAUSSIAN q-binomial COEFFICIENTS WITH RATIONAL WEIGHT FUNCTIONS

EVALUATION OF SUMS INVOLVING GAUSSIAN q-binomial COEFFICIENTS WITH RATIONAL WEIGHT FUNCTIONS EVALUATION OF SUMS INVOLVING GAUSSIAN -BINOMIAL COEFFICIENTS WITH RATIONAL WEIGHT FUNCTIONS EMRAH KILIÇ AND HELMUT PRODINGER Abstact We coside sums of the Gaussia -biomial coefficiets with a paametic atioal

More information

Lacunary Almost Summability in Certain Linear Topological Spaces

Lacunary Almost Summability in Certain Linear Topological Spaces BULLETIN of te MLYSİN MTHEMTİCL SCİENCES SOCİETY Bull. Malays. Mat. Sci. Soc. (2) 27 (2004), 27 223 Lacuay lost Suability i Cetai Liea Topological Spaces BÜNYMIN YDIN Cuuiyet Uivesity, Facutly of Educatio,

More information

Complementary Dual Subfield Linear Codes Over Finite Fields

Complementary Dual Subfield Linear Codes Over Finite Fields 1 Complemetay Dual Subfield Liea Codes Ove Fiite Fields Kiagai Booiyoma ad Somphog Jitma,1 Depatmet of Mathematics, Faculty of Sciece, Silpao Uivesity, Naho Pathom 73000, hailad e-mail : ai_b_555@hotmail.com

More information

Mapping Radius of Regular Function and Center of Convex Region. Duan Wenxi

Mapping Radius of Regular Function and Center of Convex Region. Duan Wenxi d Iteatioal Cofeece o Electical Compute Egieeig ad Electoics (ICECEE 5 Mappig adius of egula Fuctio ad Cete of Covex egio Dua Wexi School of Applied Mathematics Beijig Nomal Uivesity Zhuhai Chia 363463@qqcom

More information

IDENTITIES FOR THE NUMBER OF STANDARD YOUNG TABLEAUX IN SOME (k, l)-hooks

IDENTITIES FOR THE NUMBER OF STANDARD YOUNG TABLEAUX IN SOME (k, l)-hooks Sémiaie Lothaigie de Combiatoie 63 (010), Aticle B63c IDENTITIES FOR THE NUMBER OF STANDARD YOUNG TABLEAUX IN SOME (k, l)-hooks A. REGEV Abstact. Closed fomulas ae kow fo S(k,0;), the umbe of stadad Youg

More information

A GENERALIZATION OF A CONJECTURE OF MELHAM. 1. Introduction The Fibonomial coefficient is, for n m 1, defined by

A GENERALIZATION OF A CONJECTURE OF MELHAM. 1. Introduction The Fibonomial coefficient is, for n m 1, defined by A GENERALIZATION OF A CONJECTURE OF MELHAM EMRAH KILIC 1, ILKER AKKUS, AND HELMUT PRODINGER 3 Abstact A genealization of one of Melha s conectues is pesented Afte witing it in tes of Gaussian binoial coefficients,

More information

Advanced Physical Geodesy

Advanced Physical Geodesy Supplemetal Notes Review of g Tems i Moitz s Aalytic Cotiuatio Method. Advaced hysical Geodesy GS887 Chistophe Jekeli Geodetic Sciece The Ohio State Uivesity 5 South Oval Mall Columbus, OH 4 7 The followig

More information

Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,

Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : , MB BINOMIAL THEOREM Biomial Epessio : A algebaic epessio which cotais two dissimila tems is called biomial epessio Fo eample :,,, etc / ( ) Statemet of Biomial theoem : If, R ad N, the : ( + ) = a b +

More information

4. PERMUTATIONS AND COMBINATIONS

4. PERMUTATIONS AND COMBINATIONS 4. PERMUTATIONS AND COMBINATIONS PREVIOUS EAMCET BITS 1. The umbe of ways i which 13 gold cois ca be distibuted amog thee pesos such that each oe gets at least two gold cois is [EAMCET-000] 1) 3 ) 4 3)

More information

Using Counting Techniques to Determine Probabilities

Using Counting Techniques to Determine Probabilities Kowledge ticle: obability ad Statistics Usig outig Techiques to Detemie obabilities Tee Diagams ad the Fudametal outig iciple impotat aspect of pobability theoy is the ability to detemie the total umbe

More information

Minimal order perfect functional observers for singular linear systems

Minimal order perfect functional observers for singular linear systems Miimal ode efect fuctioal obseves fo sigula liea systems Tadeusz aczoek Istitute of Cotol Idustial lectoics Wasaw Uivesity of Techology, -66 Waszawa, oszykowa 75, POLAND Abstact. A ew method fo desigig

More information

Conditional Convergence of Infinite Products

Conditional Convergence of Infinite Products Coditioal Covegece of Ifiite Poducts William F. Tech Ameica Mathematical Mothly 106 1999), 646-651 I this aticle we evisit the classical subject of ifiite poducts. Fo stadad defiitios ad theoems o this

More information

Double Derangement Permutations

Double Derangement Permutations Ope Joural of iscrete Matheatics, 206, 6, 99-04 Published Olie April 206 i SciRes http://wwwscirporg/joural/ojd http://dxdoiorg/04236/ojd2066200 ouble erageet Perutatios Pooya aeshad, Kayar Mirzavaziri

More information

A two-sided Iterative Method for Solving

A two-sided Iterative Method for Solving NTERNATONAL JOURNAL OF MATHEMATCS AND COMPUTERS N SMULATON Volume 9 0 A two-sided teative Method fo Solvig * A Noliea Matix Equatio X= AX A Saa'a A Zaea Abstact A efficiet ad umeical algoithm is suggested

More information

Disjoint Sets { 9} { 1} { 11} Disjoint Sets (cont) Operations. Disjoint Sets (cont) Disjoint Sets (cont) n elements

Disjoint Sets { 9} { 1} { 11} Disjoint Sets (cont) Operations. Disjoint Sets (cont) Disjoint Sets (cont) n elements Disjoit Sets elemets { x, x, } X =, K Opeatios x Patitioed ito k sets (disjoit sets S, S,, K Fid-Set(x - etu set cotaiig x Uio(x,y - make a ew set by combiig the sets cotaiig x ad y (destoyig them S k

More information

Lacunary Weak I-Statistical Convergence

Lacunary Weak I-Statistical Convergence Ge. Mat. Notes, Vol. 8, No., May 05, pp. 50-58 ISSN 9-784; Copyigt ICSRS Publicatio, 05 www.i-css.og vailable ee olie at ttp//www.gema.i Lacuay Wea I-Statistical Covegece Haize Gümüş Faculty o Eegli Educatio,

More information

Application of Poisson Integral Formula on Solving Some Definite Integrals

Application of Poisson Integral Formula on Solving Some Definite Integrals Jounal of Copute and Electonic Sciences Available online at jcesblue-apog 015 JCES Jounal Vol 1(), pp 4-47, 30 Apil, 015 Application of Poisson Integal Foula on Solving Soe Definite Integals Chii-Huei

More information

a) The average (mean) of the two fractions is halfway between them: b) The answer is yes. Assume without loss of generality that p < r.

a) The average (mean) of the two fractions is halfway between them: b) The answer is yes. Assume without loss of generality that p < r. Solutios to MAML Olympiad Level 00. Factioated a) The aveage (mea) of the two factios is halfway betwee them: p ps+ q ps+ q + q s qs qs b) The aswe is yes. Assume without loss of geeality that p

More information

62. 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 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 information

6.4 Binomial Coefficients

6.4 Binomial Coefficients 64 Bioial Coefficiets Pascal s Forula Pascal s forula, aed after the seveteeth-cetury Frech atheaticia ad philosopher Blaise Pascal, is oe of the ost faous ad useful i cobiatorics (which is the foral ter

More information

«A first lesson on Mathematical Induction»

«A first lesson on Mathematical Induction» Bcgou ifotio: «A fist lesso o Mtheticl Iuctio» Mtheticl iuctio is topic i H level Mthetics It is useful i Mtheticl copetitios t ll levels It hs bee coo sight tht stuets c out the poof b theticl iuctio,

More information

On the maximum of r-stirling numbers

On the maximum of r-stirling numbers Advaces i Applied Mathematics 4 2008) 293 306 www.elsevie.com/locate/yaama O the maximum of -Stilig umbes Istvá Mező Depatmet of Algeba ad Numbe Theoy, Istitute of Mathematics, Uivesity of Debece, Hugay

More information

A GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS. Mircea Merca

A GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS. Mircea Merca Idia J Pure Appl Math 45): 75-89 February 204 c Idia Natioal Sciece Academy A GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS Mircea Merca Departmet of Mathematics Uiversity

More information

Advances in Mathematics and Statistical Sciences. On Positive Definite Solution of the Nonlinear Matrix Equation

Advances in Mathematics and Statistical Sciences. On Positive Definite Solution of the Nonlinear Matrix Equation Advace i Mathematic ad Statitical Sciece O Poitive Defiite Solutio of the Noliea * Matix Equatio A A I SANA'A A. ZAREA Mathematical Sciece Depatmet Pice Nouah Bit Abdul Rahma Uiveity B.O.Box 9Riyad 6 SAUDI

More information

MATHS FOR ENGINEERS ALGEBRA TUTORIAL 8 MATHEMATICAL PROGRESSIONS AND SERIES

MATHS FOR ENGINEERS ALGEBRA TUTORIAL 8 MATHEMATICAL PROGRESSIONS AND SERIES MATHS FOR ENGINEERS ALGEBRA TUTORIAL 8 MATHEMATICAL PROGRESSIONS AND SERIES O completio of this ttoial yo shold be able to do the followig. Eplai aithmetical ad geometic pogessios. Eplai factoial otatio

More information

Analytic Continuation

Analytic Continuation Aalytic Cotiuatio The stadard example of this is give by Example Let h (z) = 1 + z + z 2 + z 3 +... kow to coverge oly for z < 1. I fact h (z) = 1/ (1 z) for such z. Yet H (z) = 1/ (1 z) is defied for

More information

Complex Analysis Spring 2001 Homework I Solution

Complex 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 information

Automated Proofs for Some Stirling Number Identities

Automated Proofs for Some Stirling Number Identities Autoated Proofs for Soe Stirlig Nuber Idetities Mauel Kauers ad Carste Scheider Research Istitute for Sybolic Coputatio Johaes Kepler Uiversity Altebergerstraße 69 A4040 Liz, Austria Subitted: Sep 1, 2007;

More information

Fibonacci and Some of His Relations Anthony Sofo School of Computer Science and Mathematics, Victoria University of Technology,Victoria, Australia.

Fibonacci and Some of His Relations Anthony Sofo School of Computer Science and Mathematics, Victoria University of Technology,Victoria, Australia. The Mathematics Educatio ito the st Cetuy Poect Poceedigs of the Iteatioal Cofeece The Decidable ad the Udecidable i Mathematics Educatio Bo, Czech Republic, Septembe Fiboacci ad Some of His Relatios Athoy

More information

Announcements: The Rydberg formula describes. A Hydrogen-like ion is an ion that

Announcements: The Rydberg formula describes. A Hydrogen-like ion is an ion that Q: A Hydogelike io is a io that The Boh odel A) is cheically vey siila to Hydoge ios B) has the sae optical spectu as Hydoge C) has the sae ube of potos as Hydoge ) has the sae ube of electos as a Hydoge

More information

Some Remarks on the Boundary Behaviors of the Hardy Spaces

Some Remarks on the Boundary Behaviors of the Hardy Spaces Soe Reaks on the Bounday Behavios of the Hady Spaces Tao Qian and Jinxun Wang In eoy of Jaie Kelle Abstact. Soe estiates and bounday popeties fo functions in the Hady spaces ae given. Matheatics Subject

More information

EDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 2- ALGEBRAIC TECHNIQUES TUTORIAL 1 - PROGRESSIONS

EDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 2- ALGEBRAIC TECHNIQUES TUTORIAL 1 - PROGRESSIONS EDEXCEL NATIONAL CERTIFICATE UNIT 8 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME - ALGEBRAIC TECHNIQUES TUTORIAL - PROGRESSIONS CONTENTS Be able to apply algebaic techiques Aithmetic pogessio (AP): fist

More information

Research Article On Simultaneous Approximation of Modified Baskakov-Durrmeyer Operators

Research Article On Simultaneous Approximation of Modified Baskakov-Durrmeyer Operators Iteatioal Joual of Aalysis Volue 215, Aticle ID 85395, 1 pages http://dx.doi.og/1.1155/215/85395 Reseach Aticle O Siultaeous Appoxiatio of Modified Baskakov-Dueye Opeatos Pashatkua G. Patel 1,2 ad Vishu

More information

On the Combinatorics of Rooted Binary Phylogenetic Trees

On the Combinatorics of Rooted Binary Phylogenetic Trees O the Combiatoics of Rooted Biay Phylogeetic Tees Yu S. Sog Apil 3, 2003 AMS Subject Classificatio: 05C05, 92D15 Abstact We study subtee-pue-ad-egaft (SPR) opeatios o leaf-labelled ooted biay tees, also

More information

JORDAN CANONICAL FORM AND ITS APPLICATIONS

JORDAN CANONICAL FORM AND ITS APPLICATIONS JORDAN CANONICAL FORM AND ITS APPLICATIONS Shivani Gupta 1, Kaajot Kau 2 1,2 Matheatics Depatent, Khalsa College Fo Woen, Ludhiana (India) ABSTRACT This pape gives a basic notion to the Jodan canonical

More information

1. By using truth tables prove that, for all statements P and Q, the statement

1. By using truth tables prove that, for all statements P and Q, the statement Author: Satiago Salazar Problems I: Mathematical Statemets ad Proofs. By usig truth tables prove that, for all statemets P ad Q, the statemet P Q ad its cotrapositive ot Q (ot P) are equivalet. I example.2.3

More information

GRAVITATIONAL FORCE IN HYDROGEN ATOM

GRAVITATIONAL FORCE IN HYDROGEN ATOM Fudametal Joual of Mode Physics Vol. 8, Issue, 015, Pages 141-145 Published olie at http://www.fdit.com/ GRAVITATIONAL FORCE IN HYDROGEN ATOM Uiesitas Pedidika Idoesia Jl DR Setyabudhi No. 9 Badug Idoesia

More information

Counting Well-Formed Parenthesizations Easily

Counting Well-Formed Parenthesizations Easily Coutig Well-Formed Parethesizatios Easily Pekka Kilpeläie Uiversity of Easter Filad School of Computig, Kuopio August 20, 2014 Abstract It is well kow that there is a oe-to-oe correspodece betwee ordered

More information

LESSON 15: COMPOUND INTEREST

LESSON 15: COMPOUND INTEREST High School: Expoeial Fuctios LESSON 15: COMPOUND INTEREST 1. You have see this fomula fo compoud ieest. Paamete P is the picipal amou (the moey you stat with). Paamete is the ieest ate pe yea expessed

More information

Section 5.1 The Basics of Counting

Section 5.1 The Basics of Counting 1 Sectio 5.1 The Basics of Coutig Combiatorics, the study of arragemets of objects, is a importat part of discrete mathematics. I this chapter, we will lear basic techiques of coutig which has a lot of

More information

Weighted Hardy-Sobolev Type Inequality for Generalized Baouendi-Grushin Vector Fields and Its Application

Weighted Hardy-Sobolev Type Inequality for Generalized Baouendi-Grushin Vector Fields and Its Application 44Æ 3 «Vol.44 No.3 05 5 ADVANCES IN MATHEMATICS(CHINA) May 05 doi: 0.845/sxjz.03075b Weighted Hady-Sobolev Type Ieuality fo Geealized Baouedi-Gushi Vecto Fields ad Its Applicatio ZHANG Shutao HAN Yazhou

More information

SVD ( ) Linear Algebra for. A bit of repetition. Lecture: 8. Let s try the factorization. Is there a generalization? = Q2Λ2Q (spectral theorem!

SVD ( ) Linear Algebra for. A bit of repetition. Lecture: 8. Let s try the factorization. Is there a generalization? = Q2Λ2Q (spectral theorem! Liea Algeba fo Wieless Commuicatios Lectue: 8 Sigula Value Decompositio SVD Ove Edfos Depatmet of Electical ad Ifomatio echology Lud Uivesity it 00-04-06 Ove Edfos A bit of epetitio A vey useful matix

More information

A Pair of Operator Summation Formulas and Their Applications

A Pair of Operator Summation Formulas and Their Applications A Pair of Operator Suatio Forulas ad Their Applicatios Tia-Xiao He 1, Leetsch C. Hsu, ad Dogsheg Yi 3 1 Departet of Matheatics ad Coputer Sciece Illiois Wesleya Uiversity Blooigto, IL 6170-900, USA Departet

More information

Various applications of the (exponential) complete Bell polynomials. Donal F. Connon. 16 January 2010

Various applications of the (exponential) complete Bell polynomials. Donal F. Connon. 16 January 2010 Abstact Vaious applicatios of the (epoetial coplete Bell polyoials Doal F. Coo 6 Jauay I a athe staightfowa ae, we evelop the well-ow foula fo the Stilig ubes of the fist i i tes of the (epoetial coplete

More information

On Algorithm for the Minimum Spanning Trees Problem with Diameter Bounded Below

On Algorithm for the Minimum Spanning Trees Problem with Diameter Bounded Below O Algorithm for the Miimum Spaig Trees Problem with Diameter Bouded Below Edward Kh. Gimadi 1,2, Alexey M. Istomi 1, ad Ekateria Yu. Shi 2 1 Sobolev Istitute of Mathematics, 4 Acad. Koptyug aveue, 630090

More information

Solving Fuzzy Differential Equations using Runge-Kutta third order method with modified contra-harmonic mean weights

Solving Fuzzy Differential Equations using Runge-Kutta third order method with modified contra-harmonic mean weights Iteatioal Joual of Egieeig Reseach ad Geeal Sciece Volume 4, Issue 1, Jauay-Febuay, 16 Solvig Fuzzy Diffeetial Equatios usig Ruge-Kutta thid ode method with modified cota-hamoic mea weights D.Paul Dhayabaa,

More information

PUTNAM TRAINING INEQUALITIES

PUTNAM TRAINING INEQUALITIES PUTNAM TRAINING INEQUALITIES (Last updated: December, 207) Remark This is a list of exercises o iequalities Miguel A Lerma Exercises If a, b, c > 0, prove that (a 2 b + b 2 c + c 2 a)(ab 2 + bc 2 + ca

More information

On Almost Increasing Sequences For Generalized Absolute Summability

On Almost Increasing Sequences For Generalized Absolute Summability Joul of Applied Mthetic & Bioifotic, ol., o., 0, 43-50 ISSN: 79-660 (pit), 79-6939 (olie) Itetiol Scietific Pe, 0 O Alot Iceig Sequece Fo Geelized Abolute Subility W.. Suli Abtct A geel eult coceig bolute

More information

INEQUALITIES BJORN POONEN

INEQUALITIES BJORN POONEN INEQUALITIES BJORN POONEN 1 The AM-GM iequality The most basic arithmetic mea-geometric mea (AM-GM) iequality states simply that if x ad y are oegative real umbers, the (x + y)/2 xy, with equality if ad

More information

EXTENDED POWER LINDLEY DISTRIBUTION: A NEW STATISTICAL MODEL FOR NON-MONOTONE SURVIVAL DATA

EXTENDED POWER LINDLEY DISTRIBUTION: A NEW STATISTICAL MODEL FOR NON-MONOTONE SURVIVAL DATA Euopea Joual of Statistics ad Pobability Vol.3, No.3, pp.9-34, Septembe 05 Published by Euopea Cete fo Reseach Taiig ad Developmet UK (www.eajouals.og) ETENDED POWER LINDLEY DISTRIBUTION: A NEW STATISTICAL

More information

ECE 901 Lecture 4: Estimation of Lipschitz smooth functions

ECE 901 Lecture 4: Estimation of Lipschitz smooth functions ECE 9 Lecture 4: Estiatio of Lipschitz sooth fuctios R. Nowak 5/7/29 Cosider the followig settig. Let Y f (X) + W, where X is a rado variable (r.v.) o X [, ], W is a r.v. o Y R, idepedet of X ad satisfyig

More information

Riemann Hypothesis Proof

Riemann Hypothesis Proof Riea Hypothesis Proof H. Vic Dao 43 rd West Coast Nuber Theory Coferece, Dec 6-0, 009, Asiloar, CA. Revised 0/8/00. Riea Hypothesis Proof H. Vic Dao vic0@cocast.et March, 009 Revised Deceber, 009 Abstract

More information

In this simple case, the solution u = ax + b is found by integrating twice. n = 2: u = 2 u

In this simple case, the solution u = ax + b is found by integrating twice. n = 2: u = 2 u The Laplace Equatio Ei Pease I. Itoductio Defiitio. Amog the most impotat ad ubiquitous of all patial diffeetial equatios is Laplace s Equatio: u = 0, whee the Laplacia opeato acts o the fuctio u : R (

More information

30 The Electric Field Due to a Continuous Distribution of Charge on a Line

30 The Electric Field Due to a Continuous Distribution of Charge on a Line hapte 0 The Electic Field Due to a ontinuous Distibution of hage on a Line 0 The Electic Field Due to a ontinuous Distibution of hage on a Line Evey integal ust include a diffeential (such as d, dt, dq,

More information

Acta Scientiarum. Technology ISSN: Universidade Estadual de Maringá Brasil

Acta Scientiarum. Technology ISSN: Universidade Estadual de Maringá Brasil Acta cietiau Techology IN: 806-2563 edue@ueb Uiveidade Etadual de Maigá Bail Dutta, Hee; uede Reddy, Boa; Hazah Jebil, Iqbal O two ew type of tatitical covegece ad a uability ethod Acta cietiau Techology,

More information

ONE MODULO THREE GEOMETRIC MEAN LABELING OF SOME FAMILIES OF GRAPHS

ONE MODULO THREE GEOMETRIC MEAN LABELING OF SOME FAMILIES OF GRAPHS ONE MODULO THREE GEOMETRIC MEAN LABELING OF SOME FAMILIES OF GRAPHS A.Maheswari 1, P.Padiaraj 2 1,2 Departet of Matheatics,Kaaraj College of Egieerig ad Techology, Virudhuagar (Idia) ABSTRACT A graph G

More information

Zeros of Polynomials

Zeros of Polynomials Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree

More information

( a) ( ) 1 ( ) 2 ( ) ( ) 3 3 ( ) =!

( a) ( ) 1 ( ) 2 ( ) ( ) 3 3 ( ) =! .8,.9: Taylor ad Maclauri Series.8. Although we were able to fid power series represetatios for a limited group of fuctios i the previous sectio, it is ot immediately obvious whether ay give fuctio has

More information

Combinatorial Numbers and Associated Identities: Table 1: Stirling Numbers

Combinatorial Numbers and Associated Identities: Table 1: Stirling Numbers Combiatoial Numbes ad Associated Idetities: Table : Stilig Numbes Fom the seve upublished mauscipts of H. W. Gould Edited ad Compiled by Jocely Quaitace May 3, 200 Notatioal Covetios fo Table Thoughout

More information

Generating Functions and Their Applications

Generating Functions and Their Applications Geeratig Fuctios ad Their Applicatios Agustius Peter Sahaggau MIT Matheatics Departet Class of 2007 18.104 Ter Paper Fall 2006 Abstract. Geeratig fuctios have useful applicatios i ay fields of study. I

More information

Chapter 8 Complex Numbers

Chapter 8 Complex Numbers Chapte 8 Complex Numbes Motivatio: The ae used i a umbe of diffeet scietific aeas icludig: sigal aalsis, quatum mechaics, elativit, fluid damics, civil egieeig, ad chaos theo (factals. 8.1 Cocepts - Defiitio

More information

TEACHER CERTIFICATION STUDY GUIDE

TEACHER CERTIFICATION STUDY GUIDE COMPETENCY 1. ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Kow why the real ad complex umbers are each a field, ad that particular rigs are ot fields (e.g., itegers, polyomial rigs, matrix rigs) Algebra

More information

Compositions of Fuzzy T -Ideals in Ternary -Semi ring

Compositions of Fuzzy T -Ideals in Ternary -Semi ring Iteratioal Joural of Advaced i Maageet, Techology ad Egieerig Scieces Copositios of Fuy T -Ideals i Terary -Sei rig RevathiK, 2, SudarayyaP 3, Madhusudhaa RaoD 4, Siva PrasadP 5 Research Scholar, Departet

More information

Fitting the Generalized Logistic Distribution. by LQ-Moments

Fitting the Generalized Logistic Distribution. by LQ-Moments Applied Mathematical Scieces, Vol. 5, 0, o. 54, 66-676 Fittig the Geealized Logistic Distibutio by LQ-Momets Ai Shabi Depatmet of Mathematic, Uivesiti Teologi Malaysia ai@utm.my Abdul Aziz Jemai Scieces

More information

Infinite Sequences and Series

Infinite 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 information

REVISTA INVESTIGACION OPERACIONAL VOL. 38, NO.3, , 2017

REVISTA INVESTIGACION OPERACIONAL VOL. 38, NO.3, , 2017 REVISTA INVESTIGACION OPERACIONAL VOL. 8, NO., 66-71, 017 RATIO AND PRODUCT TYPE EXPONENTIAL ESTIMATORS FOR POPULATION MEAN USING RANKED SET SAMPLING Gajeda K. Vishwakaa 1*, Sayed Mohaed Zeesha *, Calos

More information

Modelling rheological cone-plate test conditions

Modelling rheological cone-plate test conditions ANNUAL TRANSACTIONS OF THE NORDIC RHEOLOGY SOCIETY, VOL. 16, 28 Modellig heological coe-plate test coditios Reida Bafod Schülle 1 ad Calos Salas-Bigas 2 1 Depatmet of Chemisty, Biotechology ad Food Sciece,

More information

Sequences and Limits

Sequences and Limits Chapter Sequeces ad Limits Let { a } be a sequece of real or complex umbers A ecessary ad sufficiet coditio for the sequece to coverge is that for ay ɛ > 0 there exists a iteger N > 0 such that a p a q

More information

PRACTICE FINAL/STUDY GUIDE SOLUTIONS

PRACTICE FINAL/STUDY GUIDE SOLUTIONS Last edited December 9, 03 at 4:33pm) Feel free to sed me ay feedback, icludig commets, typos, ad mathematical errors Problem Give the precise meaig of the followig statemets i) a f) L ii) a + f) L iii)

More information

A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence

A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as

More information

Integer Linear Programming

Integer Linear Programming Iteger Liear Programmig Itroductio Iteger L P problem (P) Mi = s. t. a = b i =,, m = i i 0, iteger =,, c Eemple Mi z = 5 s. t. + 0 0, 0, iteger F(P) = feasible domai of P Itroductio Iteger L P problem

More information

Math 2784 (or 2794W) University of Connecticut

Math 2784 (or 2794W) University of Connecticut ORDERS OF GROWTH PAT SMITH Math 2784 (or 2794W) Uiversity of Coecticut Date: Mar. 2, 22. ORDERS OF GROWTH. Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really

More information

A Class of Lindley and Weibull Distributions

A Class of Lindley and Weibull Distributions Ope Joual of Statistics 06 6 685-700 Published Olie August 06 i SciRes. http://www.scip.og/joual/ojs http://dx.doi.og/0.436/ojs.06.64058 A Class of Lidley ad Weibull Distibutios Said Hofa Alkai Depatmet

More information

[Dhayabaran*, 5(1): January, 2016] ISSN: (I2OR), Publication Impact Factor: 3.785

[Dhayabaran*, 5(1): January, 2016] ISSN: (I2OR), Publication Impact Factor: 3.785 [Dhayabaa* 5(): Jauay 206] ISSN: 2277-9655 (I2OR) Publicatio Impact Facto: 3.785 IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY SOLVING FUZZY DIFFERENTIAL EQUATIONS USING RUNGE-KUTTA

More information

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

More information

Generalized Fibonacci-Like Sequence and. Fibonacci Sequence

Generalized Fibonacci-Like Sequence and. Fibonacci Sequence It. J. Cotep. Math. Scieces, Vol., 04, o., - 4 HIKARI Ltd, www.-hiari.co http://dx.doi.org/0.88/ijcs.04.48 Geeralized Fiboacci-Lie Sequece ad Fiboacci Sequece Sajay Hare epartet of Matheatics Govt. Holar

More information

Generalized Fixed Point Theorem. in Three Metric Spaces

Generalized Fixed Point Theorem. in Three Metric Spaces It. Joural of Math. Aalysis, Vol. 4, 00, o. 40, 995-004 Geeralized Fixed Poit Thee i Three Metric Spaces Kristaq Kikia ad Luljeta Kikia Departet of Matheatics ad Coputer Scieces Faculty of Natural Scieces,

More information

Chapter 3: Theory of Modular Arithmetic 38

Chapter 3: Theory of Modular Arithmetic 38 Chapte 3: Theoy of Modula Aithmetic 38 Section D Chinese Remainde Theoem By the end of this section you will be able to pove the Chinese Remainde Theoem apply this theoem to solve simultaneous linea conguences

More information

Seunghee Ye Ma 8: Week 5 Oct 28

Seunghee Ye Ma 8: Week 5 Oct 28 Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value

More information

On the Basis Property of Eigenfunction. of the Frankl Problem with Nonlocal Parity Conditions. of the Third Kind

On the Basis Property of Eigenfunction. of the Frankl Problem with Nonlocal Parity Conditions. of the Third Kind It.. Cotemp. Math. Scieces Vol. 9 o. 3 33-38 HIKARI Lt www.m-hikai.com http://x.oi.og/.988/ijcms..33 O the Basis Popety o Eigeuctio o the Fakl Poblem with Nolocal Paity Coitios o the Thi Ki A. Sameipou

More information

Differentiable Convex Functions

Differentiable Convex Functions Differetiable Covex Fuctios The followig picture motivates Theorem 11. f ( x) f ( x) f '( x)( x x) ˆx x 1 Theorem 11 : Let f : R R be differetiable. The, f is covex o the covex set C R if, ad oly if for

More information

Sequences of Definite Integrals, Factorials and Double Factorials

Sequences 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 information

Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series

Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series

More information

x !1! + 1!2!

x !1! + 1!2! 4 Euler-Maclauri Suatio Forula 4. Beroulli Nuber & Beroulli Polyoial 4.. Defiitio of Beroulli Nuber Beroulli ubers B (,,3,) are defied as coefficiets of the followig equatio. x e x - B x! 4.. Expreesio

More information

Differentiable Convex Functions

Differentiable Convex Functions Differetiable Covex Fuctios The followig picture motivates Theorem 11. f ( x) f ( x) f '( x)( x x) ˆx x 1 Theorem 11 : Let f : R R be differetiable. The, f is covex o the covex set C R if, ad oly if for

More information

12.6 Sequential LMMSE Estimation

12.6 Sequential LMMSE Estimation 12.6 Sequetial LMMSE Estimatio Same kid if settig as fo Sequetial LS Fied umbe of paametes (but hee they ae modeled as adom) Iceasig umbe of data samples Data Model: [ H[ θ + w[ (+1) 1 p 1 [ [[0] [] ukow

More information

Sequences I. Chapter Introduction

Sequences I. Chapter Introduction Chapter 2 Sequeces I 2. Itroductio A sequece is a list of umbers i a defiite order so that we kow which umber is i the first place, which umber is i the secod place ad, for ay atural umber, we kow which

More information

IP Reference guide for integer programming formulations.

IP Reference guide for integer programming formulations. IP Referece guide for iteger programmig formulatios. by James B. Orli for 15.053 ad 15.058 This documet is iteded as a compact (or relatively compact) guide to the formulatio of iteger programs. For more

More information

Beyond simple iteration of a single function, or even a finite sequence of functions, results

Beyond simple iteration of a single function, or even a finite sequence of functions, results A Primer o the Elemetary Theory of Ifiite Compositios of Complex Fuctios Joh Gill Sprig 07 Abstract: Elemetary meas ot requirig the complex fuctios be holomorphic Theorem proofs are fairly simple ad are

More information

An analog of the arithmetic triangle obtained by replacing the products by the least common multiples

An analog of the arithmetic triangle obtained by replacing the products by the least common multiples arxiv:10021383v2 [mathnt] 9 Feb 2010 A aalog of the arithmetic triagle obtaied by replacig the products by the least commo multiples Bair FARHI bairfarhi@gmailcom MSC: 11A05 Keywords: Al-Karaji s triagle;

More information

Recurrence Relations

Recurrence Relations Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The

More information

Fundamental Theorem of Algebra. Yvonne Lai March 2010

Fundamental Theorem of Algebra. Yvonne Lai March 2010 Fudametal Theorem of Algebra Yvoe Lai March 010 We prove the Fudametal Theorem of Algebra: Fudametal Theorem of Algebra. Let f be a o-costat polyomial with real coefficiets. The f has at least oe complex

More information

Solutions to Final Exam Review Problems

Solutions to Final Exam Review Problems . Let f(x) 4+x. Solutios to Fial Exam Review Problems Math 5C, Witer 2007 (a) Fid the Maclauri series for f(x), ad compute its radius of covergece. Solutio. f(x) 4( ( x/4)) ( x/4) ( ) 4 4 + x. Sice the

More information

Induction: Solutions

Induction: Solutions Writig Proofs Misha Lavrov Iductio: Solutios Wester PA ARML Practice March 6, 206. Prove that a 2 2 chessboard with ay oe square removed ca always be covered by shaped tiles. Solutio : We iduct o. For

More information

Math 2112 Solutions Assignment 5

Math 2112 Solutions Assignment 5 Math 2112 Solutios Assigmet 5 5.1.1 Idicate which of the followig relatioships are true ad which are false: a. Z Q b. R Q c. Q Z d. Z Z Z e. Q R Q f. Q Z Q g. Z R Z h. Z Q Z a. True. Every positive iteger

More information

MA131 - Analysis 1. Workbook 2 Sequences I

MA131 - Analysis 1. Workbook 2 Sequences I MA3 - Aalysis Workbook 2 Sequeces I Autum 203 Cotets 2 Sequeces I 2. Itroductio.............................. 2.2 Icreasig ad Decreasig Sequeces................ 2 2.3 Bouded Sequeces..........................

More information