Zero Divisor Graph on Modules
|
|
- Noah Newton
- 5 years ago
- Views:
Transcription
1 J. Appl. Envron. Bol. Sc., 5S)7-3, 05 05, Textod Publcton ISSN: Journl of Appled Envronmentl nd Bologcl Scences Zero Dvsor Grph on odules Shbn Sedgh, rym Yzdn, Yhy Shbnpour 3. Deprtment of themtcs, Fculty ember, Qemshhr Brnch, Islmc Azd Unversty, Qemshhr, Irn.. Deprtment of themtcs, Fculty ember, Qemshhr Brnch, Islmc Azd Unversty, Qemshhr, Irn. 3. Deprtment of themtcs, Tbr Unversty of Bbol, Irn. eceved: y 4, 05 Accepted: August 7, 05 ABSTACT Suppose s dsplcement nd unt rng nd s module. In ths rtcle, the grph depends on, we show wth Γ, so tht, then Γ s clssc zero dvsor grph. We show tht the Γ grph wth dm Γ ) 3 s coected grph nd n Lesser module wth condton Z /{ O}, hve gr Γ ), f nd only f Γ s str grph. KEYWODS: odule, Zero dvsor grph of modulus, ound, Dmeter, Complete bprtte grph. INTODUCTION AND PELIINAIES The frst tme n 988, Beck [0] stted the concept of zero dvsor grph for commuttve rng. Beck ws consdered ll members of the dsplcement nd unt rng s vertex of the grph nd hs mn tsk ws to fnd the necessry nd suffcent condtons for the fnteness of chromtc number of grph. Also, ccordng to defnton of0 nd,two vertex of x nd y Were djcent, f nd only f xy0. evews relted to grph colorng contnued by Anderson nd Nsr [4]. But from ths grph s not obtned nterestng results. And n ddton to were obvous propertes. For exmple, ll ts vertces were djcent to zero. ecently, the zero dvsor grph from dsplcement rng hs been extended to the grph mker bsurd del from dsplcement rng. Two dels I nd J re djcent, whenever IJ O)). In [8], the clssc zero dvsor grph hs been extended to modules on dsplcement rngs. Accordng to [], m, n re djcent, f nd only f m: nm: o tht s drect extenson of clsscl zero dvsor grph. In [4] nd [3],the uthors presented two dfferent grphs to module ccordng to the frst dulty Hom, ). Although they necessrly s not generlztons of the clsscl dvsor grph, but there re some deep mutul reltons between these two grphs nd type of ts clssc. We frst nlyzed the expresson of severl bsc defntons. Defnton -): Suppose s Abeln collectve group nd s dsplcement rng, n ths cse the clls rght module, whenever sclr multplcton of elements defned n the followng wy. : m, r) mr. So tht for ech r, r, r nd m, m, m hve:. m + m) mr+ mr. m r + r) mr + mr 3. m r r) mr) r And f s unt ndm. m, then cll s untry module. Defnton -): In the G V, E) cycle wth n length s seres ofx V dstnct vertces, s x x x... x grph V s represents the verte nd E s represents the grph edge) x x 3 nsuch n. Correspondng uthor: Shbn Sedgh, Deprtment of themtcs, Fculty ember, Qemshhr Brnch, Islmc Azd Unversty, Qemshhr, Irn. E-ml: sedgh_gh@yhoo.com 7
2 Sedgh et l., 05 Defnton -3): In the G V, E) grph the shortest pth length between two u nd v vertex show wth du, v) nd the dmeter of the grph re defned s follows: dm G) SUP{ d u, v) u, v V} Defnton -4): The shortest dstnce n grph clled bck grph tht show wth grg) symbol. For exmple, the bck cube to length 4. Defnton -5): The G V, E) grph clled bprtte whenever cn the set of v vertces prttoned nto two sets, so tht between the vertces of ny set, there s no edge. In ddton, the G bprtte grph cll complete f ny two vertces n set re not prttonng be djcent to ech other. Complete grph wth n of vertex showed wth n Defnton -6): Suppose s unt dsplcement rng nd s module. In the dependent grph on mens Γ, we sym, n re djcent f nd only f m: n: o.. n esults In ths chpter, we look to rtculte defntons, theorems, concepts nd the mn results n conjuncton wth the complete nd bprtte grph from zero dvsor grph. In the followng fgure, we show zero dvsor grph the some of the Zmodules. Exmple -) K Fgure. Z s complete grph. Accordng to the bove exmple, we know tht zero dvsor grph from 3 Therefore, t s possble for the P frst number cn be result tht the Z module Z p Z s complete grph wth Z P. Theorem -): Suppose S nd S re the two Z \{ nd Γ s complete grph. 3 Z module of smple dentcl nd p S S then Proof: Suppose tht nd re two dentcl modules. Then Γ Γ ).Wth ths fct s enough to prove the theorem. Web show tht~s S. For echo x S hve x, o) : S). Also for ech, b) hve, b) o) : o nd lso o) s djcent to ech element of, b) nonzero. Ths result s cheved to o). Now suppose tht x nd y s two nonzero element from S. It s esy to show tht x) S) s del mxml from. It s obvous tht : s ncludng S). On the other hnd f : then we hve x, o) nd lso for ech r, we hve x, o) r. Then yr 0 be result tht r nd lso x xr 0, whch s nconsstency. Therefore, : S) nd )~ s djcent ny nonzero element from. 8
3 J. Appl. Envron. Bol. Sc., 5S)7-3, 05 Theorem -): Suppose s commuttve rng. In ths cse, s feld f nd only f Γ modulus, the s complete grph. ) for ech Proof: Suppose tht s feld. If dm ) then Γ φnd therefore, Γ s complete grph. If dm ) for echo m, we hve m:, becuse there s o r m:, tht result s r m nd lso mr mtht s nconsstency. Therefore, for ech m, n element from, we hven m: o. ) The No ssumpton s del mxml from. Put. Then for ech No x / N o ndo r, we hve o) o, r) : ) o. So for ech dstnct element of s, r pposng No zero n, we hve o, r) o, s) : ) o. Then for ech o, S, we hve o, )No S o. No Becuse NoS o,s): ). And ths mples tht for ech s o,, we hve N o S o) nd N N o s) ond therefore, N o o) nd so o s feld. Theorem -4): Suppose s modulus of sem-smple fnte genertor, whch homogeneous components s smple, then y s djcent member of \{, f nd only fx I y o. Proof: Suppose tht I S, whch Sre non-dentcl smple below modules from. suppose tht y Z re djcent. We need to show thtx I y o. SupposexI y o, therefore, there s α IthtS xiy, snce y nd x re below modules from, there re subsdres of A nd B from I α tht x S ) then: nd y S ). Lemm t the source [7]). Suppose x y: o) A B y: y: y BS) BT) I B S) And x ~ I\ AS s result: x) I\ AS) I I\ A S).However, snce x y: o, therefore, y: nd lso we hve S I S ). However, snce for ech, j I S) Sj) re prelmnry, so for ech r I\ A, we hve I S ) S ) I S ) S ). However, forr I\ A, there s j r B, whch B B I\ A r S jr) Sr) S jr) Sr nd so we hve ). So S jr ~ Sr nd ccordng to our ssumpton S jr S r. Becuse, there s α I tht Sα xiy. Snce S x~ α I\ AS, therefore, there s I\ A, whch Sα ~S. However, the bove equtons, there s j j Btht S α S Sj, whch n result, we hve Sα yi BS) o), whch s n nconsstency. Proof the second sde, s resultng usng the followng lemm. Lemm -3): Suppose s module nd m nd n re n nonzero element from. ) If m nd n re djcent, then for echr,s tht mr ond ns o, we hve mr ns o. ) If m I n o, then m nd n re djcent. Proof ): Supposem n: o, esly t cn be shown tht for echr, mr n: o.on the other hnd for echs, we hve ns: n:. So mr ns: mr n: o. 3) Snce n: nim s resulted m n: otht cn be concludedm n o B I\A. 9
4 Sedgh et l., 05 esult -4): Suppose tht nd re two non-dentcl smple sub grph from. Then Γ s complete bprtte grph. Proof proof of ):Suppose We re showng tht x nd y Z re djcent, ccordng to the before theorem x y o) y j, whch non-dentcl smple sub modulus From, therefore, I. jnd, j {, }. It s esy to show tht x nd y re two x nd Second proof):for echx nd y, we hve y for ech x I y j j. o. Therefore, from lemm -5), x nd re not djcent. If y \{ so tht x y: o, : nd therefore, y re djcent. Show tht ny two elements from then x y: : o. On the other ) )) otht s resultng ) ). Becuse ) s del mxml from, so ) ), then ~, whch s contrdcton. As result, we show ) ) tht foro x nd o y, x+ re not djcent of ny element from for, Z, we hve: z : : ) therefore, x z: o x+ z: x+ ) y nd for ech + s resulted tht o )) nd so ) y )) ). Becuse ) s del mxml from. Therefore ), ths s contrdcton. If z x+ : o, then by lemm -9) n the source [7] one of the followng concluson: Cse ):If x+, then ~ ~, n the other words ~ ~, whch s not smple module, becuse: x+ I I, I, whch result x) ~, whch s contrdcton. Therefore, I I nd therefore < s contrdcton. Cse ): If x+ then smlr cse, contrdcton occurs. Cse 3): Suppose x+ therefore, x + : :, so z x+ : o s resultng tht z o tht ths s contrdcton. Smlrly, by replcng nsted, lso contrdcton occurs. Fnlly, bout the x + x + y ) : o s resultng tht n y x + y ) : o, x x + y ) : o, whch s mpossble. Suppose for n 3tht ts homogeneous components re smple. Cn be predcted n ths cse the Γ s complete n-prt grph.in theorem the source 4- [6]) hs been shown to reducton the dsplcement rng, we hve tht the non-empty Γ), n s wth gr Γ )), f nd only f Γ ) K for ech n, then, we generlze ths result for Γ. Defnton -7): The module of clled reducton, whenever for ech nd m tht we hve. m othen m o. Lemm -8): Suppose s reducton module wth Z \{. If Γ s bprtte grph by sector of V nd V then V U{ for ech, s sub module from. V Proof: Supposer, xx V. We must showv + V V nd rx V. If then V Now supposerx o. From ssumpton, xs djcent of n element from V n the nme ofy. If y rx o rx. rx then 0
5 J. Appl. Envron. Bol. Sc., 5S)7-3, 05 y y : o r y: o, o from lemm -5), tht s resultng for ech m, mr, snce the s reducton module, so mr o, whch s resultng m s djcent ytht s contrdcton. So rx y nd from lemm -5, rxs djcent y, snce y V, therefore rx V. If xor xre equl to zero, then x+ x V, so t cn be ssumed tht none of thex or xs not equl to zero. Snce tht x V, therefore, there s y, y V, whch x re djcent y, for ech,. From lemm -5, we hve y y o) w y y, scence tht for ech,, we hve x I, therefore, o I w: x w VI V o) V nd f x + x o, becuse V s belongng to s sub module of. Lemm -9): If I, we hve x + x )w: o), now f x + x o, tht w therefore, x+ x V, smlrly, cn be shown thtv m Z, then m s fundmentl sub module from. m I K o, from m Z, whch s contrdcton, therefore, Proof: If m s not fundmentl, then there s non-zero sub module of K from, whch lemm -5, the m s djcent ech nonzero element from K, so m s fundmentl sub module from. Theorem -0): Suppose s reducton module wth \{ then followng tems concluson: ) Γ s bprtte grph. ). d U n Z. If Γ s bprtte grph, Proof ): SupposeZ V UV V IV nd no element ofv re not djcent. From lemm -8, we hve: V V U{ nd V V U{ re sub modules of. For ech Z Vnd y V, we hve Z I y V IV o) Proof ): Snce V) z nd tht φ nd from lemm -5, z nd y re djcent. zv ) re empty, of lemm -9, ech submodule of V nd V re fundmentl. So, V nd V re unform sub module from. Now we showv Vn s fundmentl. Suppose K s sub module from, whch K IV V ) o) nd o) y K. Then, for ech o z V nd o z V, we hve zi y o) ziw. Therefore, z s djcent of y nd w, so tht z VI V o), whch s contrdcton, therefore, V V n s fundmentl. Lemm -): Suppose s reducton module wth Z \{. Then gr Γ ) f nd only f Γ s str grph. Proof ):It s obvous. ) Suppose Γ s not ncludng cycle, Γ s tree nd therefore s bprtte grph. Now from theorem-0, Γ s complete bprtte grph. SupposeV nd V re prt of Γ grph. Snce the Γ s not ny cycle, we hve V or V, whch concluded tht Γ s str grph. Defne -): A hlf group grph s bprtte zero dvsor, f nd only f s not ncludng ny trngle. [4]) Lemm -3): Suppose s module. Γ s ncludng cycle of odd length, then Γ s trngle. Proof: By nducton, we show for ech cycle the odd length, n + 5, there s cycle of length of k+ for k< n. Supposex x x3... x n x n+ x s cycle of odd length of n +. If two non-consecutve vote of xnd x jre djcent, then proof s complete. Otherwse element of o) z x Ix3 o) from lemm -5, z for ech n +. So gn z s djcent to both elements of xndx n+. Therefore, we x x + z x x... hve the cycle of n 4 5 n+, tht ths s desred sme dstnce. x
6 Sedgh et l., 05 Theorem -4): Suppose s module. If s gr Γ ) 4, then Γ s bprtte grph wth prts of V nd V, so tht V, V. Conversely, t s true f the s reducton module wth Z \{. Proof: Supposegr Γ ) 4. Usng lemm -3, we show tht the length of ech cycle of Γ s even. Snce Γ hs cycle of length 4, ths hypothess s confrmed. On the contrry, lso theorem -0cn s proved. In the followng, we expln the between generlzton reltonshp from the defnton of zero dvsor grphs n [], t wll show byγ b ) nd wht, we hve shown n ths pper. Frst, t s worth notng tht Γ s sub grph from Γb tht f m, n Z re two djcent vertces n Γ, Or such s equvlence n m: o or m n: othen n: m: o. However, the reverse s not true s the followng exmple. Exmple -5): Suppose Z Z4s Z module. Then Γ b Kv, however Γ dfferent fromk v, whch we re showng n fgure.. Fgure. However, when the s multplctve module As for ech sub modules of N from, there s del of I from, whch N I), then Γ Γb. Suppose m: n: o. Therefore, n : nnd m : m, so both of m n: ondn m: o. EFEENCES [] G. Alpour, S. Akbr,. Behbood,. Nkndsh, J Nkmehr, nd F. Shves, The clssfcton of the hltng-del grphs of commuttve rngs, Algebr Colloquum, to pper). [] S. Akbr nd A. ohmmdn, on zero-dvsor grphs of fnte rngs, J. Algebr ), no., [3] DF Anderson, C Axtell, nd JA Stckles, Jr., Zero-dvsor grphs n commuttve rngs, n Commuttve Algebr, Noethern nd Non-Noethern Perspectves. Fontn, S.-E. Kbbj, B. Olberdng, I. Swnson, Eds.), 3-45, Sprnger-Verlg, New York, 00. [4] DF Anderson, A. Frzer, A. Luvem nd PS Lvngston, The zero-dvsor grph of commuttve rng, II, n: Lecture Notes n Pure nd Appl. th., Vol. 0, pp. 6-7, Dekker, New York, 00. [5] DF Anderson, nd PS Lvngston, The zero-dvsor grph of commuttve rng, J. Algebr 7 999), no., [6] DF Anderson, nd PS uly, On the dmeter nd grth of zero-dvsor grph, J. Pure Appl. Algerbr 0 007), no., [7] FW Anderson nd K Fuller, ngs nd Ctegores of odules, Sprnger-Verlg, New York, 99. [8]. Bzr, E. omthn, nd S. Sfeeyn, A zero-dvsor grph for modules wth respect to ther frst) dul, J. Algebr Appl. 03), no., 5-36, pp. [9] A zero-dvsor grph for modules wth respect to elements of ther frst) dul, submtted to Bull. Of IS. [0] I. Beck, Colorng of commuttve rngs, J. Algebr 6 988), no., []. Behbood, zero dvsor grphs for modules over commuttve rngs, J. Commut, Algebr 4 0), no.,
7 J. Appl. Envron. Bol. Sc., 5S)7-3, 05 []. Behbood nd Z. kee, The hltng-del grph of commuttve rngs I, J.Algebr Appl. 0 0), no. 4, [3] , The hltng-del grph of commuttve rngs II, J. Algebr Appl. 0 0), no. 4, [4] J. Duns nd L. Fuchs, Infnte Golde dmensons, J. Algebr 988), no., [5] F. Deeyer nd K. Schneder, Auto orphsms nd zero-dvsor grphs of commuttve rngs, Internet J. Commuttve ngs 00), no. 3, [6] D. Lu nd T. Wu, On bprtte zero-dvsor grphs, Dscrete th ), no. 4, [7] SB uly, Cycles nd symmetres of zero-dvsors, Comm. Algebr 30 00), no. 7, [8] SP edmond, The zero-dvsor grph of non-commuttve rng, Internet J. Com-muttve ngs 00), no.4, 03-. [9] S. Sfeeyn,. Bzr nd E. omthn, A GENEALIZATION OF THE ZEO-DIVISO GAPH FO ODULES, J. Koren th. Soc. 5 04), No, pp [0] DB West Introducton to Grph Theory, nd ed., Prentce Hll, Upper Sddle ver, 00. []. Wsbuer, Foundtons of odules nd ng Theory, Gordon nd Brech edng 99. 3
The Number of Rows which Equal Certain Row
Interntonl Journl of Algebr, Vol 5, 011, no 30, 1481-1488 he Number of Rows whch Equl Certn Row Ahmd Hbl Deprtment of mthemtcs Fcult of Scences Dmscus unverst Dmscus, Sr hblhmd1@gmlcom Abstrct Let be X
More informationTwo Coefficients of the Dyson Product
Two Coeffcents of the Dyson Product rxv:07.460v mth.co 7 Nov 007 Lun Lv, Guoce Xn, nd Yue Zhou 3,,3 Center for Combntorcs, LPMC TJKLC Nnk Unversty, Tnjn 30007, P.R. Chn lvlun@cfc.nnk.edu.cn gn@nnk.edu.cn
More informationRank One Update And the Google Matrix by Al Bernstein Signal Science, LLC
Introducton Rnk One Updte And the Google Mtrx y Al Bernsten Sgnl Scence, LLC www.sgnlscence.net here re two dfferent wys to perform mtrx multplctons. he frst uses dot product formulton nd the second uses
More informationInternational Journal of Pure and Applied Sciences and Technology
Int. J. Pure Appl. Sc. Technol., () (), pp. 44-49 Interntonl Journl of Pure nd Appled Scences nd Technolog ISSN 9-67 Avlle onlne t www.jopst.n Reserch Pper Numercl Soluton for Non-Lner Fredholm Integrl
More informationFINITE NEUTROSOPHIC COMPLEX NUMBERS. W. B. Vasantha Kandasamy Florentin Smarandache
INITE NEUTROSOPHIC COMPLEX NUMBERS W. B. Vsnth Kndsmy lorentn Smrndche ZIP PUBLISHING Oho 11 Ths book cn be ordered from: Zp Publshng 1313 Chespeke Ave. Columbus, Oho 31, USA Toll ree: (61) 85-71 E-ml:
More information7.2 Volume. A cross section is the shape we get when cutting straight through an object.
7. Volume Let s revew the volume of smple sold, cylnder frst. Cylnder s volume=se re heght. As llustrted n Fgure (). Fgure ( nd (c) re specl cylnders. Fgure () s rght crculr cylnder. Fgure (c) s ox. A
More informationINTRODUCTION TO COMPLEX NUMBERS
INTRODUCTION TO COMPLEX NUMBERS The numers -4, -3, -, -1, 0, 1,, 3, 4 represent the negtve nd postve rel numers termed ntegers. As one frst lerns n mddle school they cn e thought of s unt dstnce spced
More informationA Family of Multivariate Abel Series Distributions. of Order k
Appled Mthemtcl Scences, Vol. 2, 2008, no. 45, 2239-2246 A Fmly of Multvrte Abel Seres Dstrbutons of Order k Rupk Gupt & Kshore K. Ds 2 Fculty of Scence & Technology, The Icf Unversty, Agrtl, Trpur, Ind
More informationMATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35
MATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35 9. Modules over PID This week we re proving the fundmentl theorem for finitely generted modules over PID, nmely tht they re ll direct sums of cyclic modules.
More informationReview of linear algebra. Nuno Vasconcelos UCSD
Revew of lner lgebr Nuno Vsconcelos UCSD Vector spces Defnton: vector spce s set H where ddton nd sclr multplcton re defned nd stsf: ) +( + ) (+ )+ 5) λ H 2) + + H 6) 3) H, + 7) λ(λ ) (λλ ) 4) H, - + 8)
More information4. Eccentric axial loading, cross-section core
. Eccentrc xl lodng, cross-secton core Introducton We re strtng to consder more generl cse when the xl force nd bxl bendng ct smultneousl n the cross-secton of the br. B vrtue of Snt-Vennt s prncple we
More informationLecture 4: Piecewise Cubic Interpolation
Lecture notes on Vrtonl nd Approxmte Methods n Appled Mthemtcs - A Perce UBC Lecture 4: Pecewse Cubc Interpolton Compled 6 August 7 In ths lecture we consder pecewse cubc nterpolton n whch cubc polynoml
More informationChapter Newton-Raphson Method of Solving a Nonlinear Equation
Chpter.4 Newton-Rphson Method of Solvng Nonlner Equton After redng ths chpter, you should be ble to:. derve the Newton-Rphson method formul,. develop the lgorthm of the Newton-Rphson method,. use the Newton-Rphson
More informationDennis Bricker, 2001 Dept of Industrial Engineering The University of Iowa. MDP: Taxi page 1
Denns Brcker, 2001 Dept of Industrl Engneerng The Unversty of Iow MDP: Tx pge 1 A tx serves three djcent towns: A, B, nd C. Ech tme the tx dschrges pssenger, the drver must choose from three possble ctons:
More informationAli Omer Alattass Department of Mathematics, Faculty of Science, Hadramout University of science and Technology, P. O. Box 50663, Mukalla, Yemen
Journal of athematcs and Statstcs 7 (): 4448, 0 ISSN 5493644 00 Scence Publcatons odules n σ[] wth Chan Condtons on Small Submodules Al Omer Alattass Department of athematcs, Faculty of Scence, Hadramout
More informationThe Schur-Cohn Algorithm
Modelng, Estmton nd Otml Flterng n Sgnl Processng Mohmed Njm Coyrght 8, ISTE Ltd. Aendx F The Schur-Cohn Algorthm In ths endx, our m s to resent the Schur-Cohn lgorthm [] whch s often used s crteron for
More informationFACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP
C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class
More informationJens Siebel (University of Applied Sciences Kaiserslautern) An Interactive Introduction to Complex Numbers
Jens Sebel (Unversty of Appled Scences Kserslutern) An Interctve Introducton to Complex Numbers 1. Introducton We know tht some polynoml equtons do not hve ny solutons on R/. Exmple 1.1: Solve x + 1= for
More informationPrinciple Component Analysis
Prncple Component Anlyss Jng Go SUNY Bufflo Why Dmensonlty Reducton? We hve too mny dmensons o reson bout or obtn nsghts from o vsulze oo much nose n the dt Need to reduce them to smller set of fctors
More informationKatholieke Universiteit Leuven Department of Computer Science
Updte Rules for Weghted Non-negtve FH*G Fctorzton Peter Peers Phlp Dutré Report CW 440, Aprl 006 Ktholeke Unverstet Leuven Deprtment of Computer Scence Celestjnenln 00A B-3001 Heverlee (Belgum) Updte Rules
More informationEffects of polarization on the reflected wave
Lecture Notes. L Ros PPLIED OPTICS Effects of polrzton on the reflected wve Ref: The Feynmn Lectures on Physcs, Vol-I, Secton 33-6 Plne of ncdence Z Plne of nterfce Fg. 1 Y Y r 1 Glss r 1 Glss Fg. Reflecton
More informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More informationFoundations of Arithmetic
Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an
More informationChapter 2 Introduction to Algebra. Dr. Chih-Peng Li ( 李 )
Chpter Introducton to Algebr Dr. Chh-Peng L 李 Outlne Groups Felds Bnry Feld Arthetc Constructon of Glos Feld Bsc Propertes of Glos Feld Coputtons Usng Glos Feld Arthetc Vector Spces Groups 3 Let G be set
More informationSequences of Intuitionistic Fuzzy Soft G-Modules
Interntonl Mthemtcl Forum, Vol 13, 2018, no 12, 537-546 HIKARI Ltd, wwwm-hkrcom https://doorg/1012988/mf201881058 Sequences of Intutonstc Fuzzy Soft G-Modules Velyev Kemle nd Huseynov Afq Bku Stte Unversty,
More informationApplied Statistics Qualifier Examination
Appled Sttstcs Qulfer Exmnton Qul_june_8 Fll 8 Instructons: () The exmnton contns 4 Questons. You re to nswer 3 out of 4 of them. () You my use ny books nd clss notes tht you mght fnd helpful n solvng
More informationDCDM BUSINESS SCHOOL NUMERICAL METHODS (COS 233-8) Solutions to Assignment 3. x f(x)
DCDM BUSINESS SCHOOL NUMEICAL METHODS (COS -8) Solutons to Assgnment Queston Consder the followng dt: 5 f() 8 7 5 () Set up dfference tble through fourth dfferences. (b) Wht s the mnmum degree tht n nterpoltng
More informationBinding Number and Connected (g, f + 1)-Factors in Graphs
Binding Number nd Connected (g, f + 1)-Fctors in Grphs Jinsheng Ci, Guizhen Liu, nd Jinfeng Hou School of Mthemtics nd system science, Shndong University, Jinn 50100, P.R.Chin helthci@163.com Abstrct.
More informationProof that if Voting is Perfect in One Dimension, then the First. Eigenvector Extracted from the Double-Centered Transformed
Proof tht f Votng s Perfect n One Dmenson, then the Frst Egenvector Extrcted from the Doule-Centered Trnsformed Agreement Score Mtrx hs the Sme Rn Orderng s the True Dt Keth T Poole Unversty of Houston
More informationThe Study of Lawson Criterion in Fusion Systems for the
Interntonl Archve of Appled Scences nd Technology Int. Arch. App. Sc. Technol; Vol 6 [] Mrch : -6 Socety of ducton, Ind [ISO9: 8 ertfed Orgnzton] www.soeg.co/st.html OD: IAASA IAAST OLI ISS - 6 PRIT ISS
More informationThe graphs of Rational Functions
Lecture 4 5A: The its of Rtionl Functions s x nd s x + The grphs of Rtionl Functions The grphs of rtionl functions hve severl differences compred to power functions. One of the differences is the behvior
More informationCOMPLEX NUMBER & QUADRATIC EQUATION
MCQ COMPLEX NUMBER & QUADRATIC EQUATION Syllus : Comple numers s ordered prs of rels, Representton of comple numers n the form + nd ther representton n plne, Argnd dgrm, lger of comple numers, modulus
More informationChapter 1: Fundamentals
Chpter 1: Fundmentls 1.1 Rel Numbers Types of Rel Numbers: Nturl Numbers: {1, 2, 3,...}; These re the counting numbers. Integers: {... 3, 2, 1, 0, 1, 2, 3,...}; These re ll the nturl numbers, their negtives,
More informationESCI 342 Atmospheric Dynamics I Lesson 1 Vectors and Vector Calculus
ESI 34 tmospherc Dnmcs I Lesson 1 Vectors nd Vector lculus Reference: Schum s Outlne Seres: Mthemtcl Hndbook of Formuls nd Tbles Suggested Redng: Mrtn Secton 1 OORDINTE SYSTEMS n orthonorml coordnte sstem
More informationChapter Newton-Raphson Method of Solving a Nonlinear Equation
Chpter 0.04 Newton-Rphson Method o Solvng Nonlner Equton Ater redng ths chpter, you should be ble to:. derve the Newton-Rphson method ormul,. develop the lgorthm o the Newton-Rphson method,. use the Newton-Rphson
More informationMath 497C Sep 17, Curves and Surfaces Fall 2004, PSU
Mth 497C Sep 17, 004 1 Curves nd Surfces Fll 004, PSU Lecture Notes 3 1.8 The generl defnton of curvture; Fox-Mlnor s Theorem Let α: [, b] R n be curve nd P = {t 0,...,t n } be prtton of [, b], then the
More informationVolume 18 Figure 1. Notation 1. Notation 2. Observation 1. Remark 1. Remark 2. Remark 3. Remark 4. Remark 5. Remark 6. Theorem A [2]. Theorem B [2].
Bulletn of Mathematcal Scences and Applcatons Submtted: 016-04-07 ISSN: 78-9634, Vol. 18, pp 1-10 Revsed: 016-09-08 do:10.1805/www.scpress.com/bmsa.18.1 Accepted: 016-10-13 017 ScPress Ltd., Swtzerland
More informationWe partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.
Mth 255 - Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn
More informationUNIVERSITY OF IOANNINA DEPARTMENT OF ECONOMICS. M.Sc. in Economics MICROECONOMIC THEORY I. Problem Set II
Mcroeconomc Theory I UNIVERSITY OF IOANNINA DEPARTMENT OF ECONOMICS MSc n Economcs MICROECONOMIC THEORY I Techng: A Lptns (Note: The number of ndctes exercse s dffculty level) ()True or flse? If V( y )
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationOn the set of natural numbers
On the set of natural numbers by Jalton C. Ferrera Copyrght 2001 Jalton da Costa Ferrera Introducton The natural numbers have been understood as fnte numbers, ths wor tres to show that the natural numbers
More informationGraph Theory. Dr. Saad El-Zanati, Faculty Mentor Ryan Bunge Graduate Assistant Illinois State University REU. Graph Theory
Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Grph Theory Dniel Gibson, Concordi University Jckelyn Ngel, Dominicn University Benjmin Stnley, New Mexico Stte University Allison
More informationOn super edge-magic total labeling of banana trees
On super edge-mgic totl lbeling of bnn trees M. Hussin 1, E. T. Bskoro 2, Slmin 3 1 School of Mthemticl Sciences, GC University, 68-B New Muslim Town, Lhore, Pkistn mhmths@yhoo.com 2 Combintoril Mthemtics
More informationChapter 3 MATRIX. In this chapter: 3.1 MATRIX NOTATION AND TERMINOLOGY
Chpter 3 MTRIX In this chpter: Definition nd terms Specil Mtrices Mtrix Opertion: Trnspose, Equlity, Sum, Difference, Sclr Multipliction, Mtrix Multipliction, Determinnt, Inverse ppliction of Mtrix in
More informationRandić Energy and Randić Estrada Index of a Graph
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 5, No., 202, 88-96 ISSN 307-5543 www.ejpam.com SPECIAL ISSUE FOR THE INTERNATIONAL CONFERENCE ON APPLIED ANALYSIS AND ALGEBRA 29 JUNE -02JULY 20, ISTANBUL
More informationTHE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens
THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of
More informationStatistical Mechanics and Combinatorics : Lecture III
Statstcal Mechancs and Combnatorcs : Lecture III Dmer Model Dmer defntons Defnton A dmer coverng (perfect matchng) of a fnte graph s a set of edges whch covers every vertex exactly once, e every vertex
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
More information12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA. 4. Tensor product
12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA Here s an outlne of what I dd: (1) categorcal defnton (2) constructon (3) lst of basc propertes (4) dstrbutve property (5) rght exactness (6) localzaton
More informationNOTE AN INEQUALITY FOR KRUSKAL-MACAULAY FUNCTIONS
NOTE AN INEQUALITY FOR KRUSKAL-MACAULAY FUNCTIONS BERNARDO M. ÁBREGO, SILVIA FERNÁNDEZ-MERCHANT, AND BERNARDO LLANO Abstrct. Gven ntegers nd n, there s unque wy of wrtng n s n = n n... n so tht n <
More informationModulo Magic Labeling in Digraphs
Gen. Math. Notes, Vol. 7, No., August, 03, pp. 5- ISSN 9-784; Copyrght ICSRS Publcaton, 03 www.-csrs.org Avalable free onlne at http://www.geman.n Modulo Magc Labelng n Dgraphs L. Shobana and J. Baskar
More informationModelli Clamfim Equazione del Calore Lezione ottobre 2014
CLAMFIM Bologna Modell 1 @ Clamfm Equazone del Calore Lezone 17 15 ottobre 2014 professor Danele Rtell danele.rtell@unbo.t 1/24? Convoluton The convoluton of two functons g(t) and f(t) s the functon (g
More informationSpanning tree congestion of some product graphs
Spnning tree congestion of some product grphs Hiu-Fi Lw Mthemticl Institute Oxford University 4-9 St Giles Oxford, OX1 3LB, United Kingdom e-mil: lwh@mths.ox.c.uk nd Mikhil I. Ostrovskii Deprtment of Mthemtics
More informationGAUSS ELIMINATION. Consider the following system of algebraic linear equations
Numercl Anlyss for Engneers Germn Jordnn Unversty GAUSS ELIMINATION Consder the followng system of lgebrc lner equtons To solve the bove system usng clsscl methods, equton () s subtrcted from equton ()
More informationTHE COMBINED SHEPARD ABEL GONCHAROV UNIVARIATE OPERATOR
REVUE D ANALYSE NUMÉRIQUE ET DE THÉORIE DE L APPROXIMATION Tome 32, N o 1, 2003, pp 11 20 THE COMBINED SHEPARD ABEL GONCHAROV UNIVARIATE OPERATOR TEODORA CĂTINAŞ Abstrct We extend the Sheprd opertor by
More informationRegular Language. Nonregular Languages The Pumping Lemma. The pumping lemma. Regular Language. The pumping lemma. Infinitely long words 3/17/15
Regulr Lnguge Nonregulr Lnguges The Pumping Lemm Models of Comput=on Chpter 10 Recll, tht ny lnguge tht cn e descried y regulr expression is clled regulr lnguge In this lecture we will prove tht not ll
More informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More informationPhysics 121 Sample Common Exam 2 Rev2 NOTE: ANSWERS ARE ON PAGE 7. Instructions:
Physcs 121 Smple Common Exm 2 Rev2 NOTE: ANSWERS ARE ON PAGE 7 Nme (Prnt): 4 Dgt ID: Secton: Instructons: Answer ll 27 multple choce questons. You my need to do some clculton. Answer ech queston on the
More informationNP-Completeness : Proofs
NP-Completeness : Proofs Proof Methods A method to show a decson problem Π NP-complete s as follows. (1) Show Π NP. (2) Choose an NP-complete problem Π. (3) Show Π Π. A method to show an optmzaton problem
More informationHMMT February 2016 February 20, 2016
HMMT February 016 February 0, 016 Combnatorcs 1. For postve ntegers n, let S n be the set of ntegers x such that n dstnct lnes, no three concurrent, can dvde a plane nto x regons (for example, S = {3,
More informationReversing the Chain Rule. As we have seen from the Second Fundamental Theorem ( 4.3), the easiest way to evaluate an integral b
Mth 32 Substitution Method Stewrt 4.5 Reversing the Chin Rule. As we hve seen from the Second Fundmentl Theorem ( 4.3), the esiest wy to evlute n integrl b f(x) dx is to find n ntiderivtive, the indefinite
More information1 Matrix representations of canonical matrices
1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:
More informationSmarandache-Zero Divisors in Group Rings
Smarandache-Zero Dvsors n Group Rngs W.B. Vasantha and Moon K. Chetry Department of Mathematcs I.I.T Madras, Chenna The study of zero-dvsors n group rngs had become nterestng problem snce 1940 wth the
More informationChapter I Vector Analysis
. Chpte I Vecto nlss . Vecto lgeb j It s well-nown tht n vecto cn be wtten s Vectos obe the followng lgebc ules: scl s ) ( j v v cos ) ( e Commuttv ) ( ssoctve C C ) ( ) ( v j ) ( ) ( ) ( ) ( (v) he lw
More informationBases for Vector Spaces
Bses for Vector Spces 2-26-25 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything
More informationInternational Journal of Algebra, Vol. 8, 2014, no. 5, HIKARI Ltd,
Internatonal Journal of Algebra, Vol. 8, 2014, no. 5, 229-238 HIKARI Ltd, www.m-hkar.com http://dx.do.org/10.12988/ja.2014.4212 On P-Duo odules Inaam ohammed Al Had Department of athematcs College of Educaton
More informationThe Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.
Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F
More informationHarvard University Computer Science 121 Midterm October 23, 2012
Hrvrd University Computer Science 121 Midterm Octoer 23, 2012 This is closed-ook exmintion. You my use ny result from lecture, Sipser, prolem sets, or section, s long s you quote it clerly. The lphet is
More informationPAijpam.eu SOME NEW SUM PERFECT SQUARE GRAPHS S.G. Sonchhatra 1, G.V. Ghodasara 2
Internatonal Journal of Pure and Appled Mathematcs Volume 113 No. 3 2017, 489-499 ISSN: 1311-8080 (prnted verson); ISSN: 1314-3395 (on-lne verson) url: http://www.jpam.eu do: 10.12732/jpam.v1133.11 PAjpam.eu
More informationOnline Appendix to. Mandating Behavioral Conformity in Social Groups with Conformist Members
Onlne Appendx to Mndtng Behvorl Conformty n Socl Groups wth Conformst Members Peter Grzl Andrze Bnk (Correspondng uthor) Deprtment of Economcs, The Wllms School, Wshngton nd Lee Unversty, Lexngton, 4450
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationQuantum Codes from Generalized Reed-Solomon Codes and Matrix-Product Codes
1 Quntum Codes from Generlzed Reed-Solomon Codes nd Mtrx-Product Codes To Zhng nd Gennn Ge Abstrct rxv:1508.00978v1 [cs.it] 5 Aug 015 One of the centrl tsks n quntum error-correcton s to construct quntum
More informationComplete subgraphs in multipartite graphs
Complete subgraphs n multpartte graphs FLORIAN PFENDER Unverstät Rostock, Insttut für Mathematk D-18057 Rostock, Germany Floran.Pfender@un-rostock.de Abstract Turán s Theorem states that every graph G
More informationThe Pseudoblocks of Endomorphism Algebras
Internatonal Mathematcal Forum, 4, 009, no. 48, 363-368 The Pseudoblocks of Endomorphsm Algebras Ahmed A. Khammash Department of Mathematcal Scences, Umm Al-Qura Unversty P.O.Box 796, Makkah, Saud Araba
More informationAdvanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
More informationStatistics and Probability Letters
Sttstcs nd Probblty Letters 79 (2009) 105 111 Contents lsts vlble t ScenceDrect Sttstcs nd Probblty Letters journl homepge: www.elsever.com/locte/stpro Lmtng behvour of movng verge processes under ϕ-mxng
More informationRemarks on the Properties of a Quasi-Fibonacci-like Polynomial Sequence
Remarks on the Propertes of a Quas-Fbonacc-lke Polynomal Sequence Brce Merwne LIU Brooklyn Ilan Wenschelbaum Wesleyan Unversty Abstract Consder the Quas-Fbonacc-lke Polynomal Sequence gven by F 0 = 1,
More informationLecture 3 ( ) (translated and slightly adapted from lecture notes by Martin Klazar)
Lecture 3 (5.3.2018) (trnslted nd slightly dpted from lecture notes by Mrtin Klzr) Riemnn integrl Now we define precisely the concept of the re, in prticulr, the re of figure U(, b, f) under the grph of
More informationThe binomial transforms of the generalized (s, t )-Jacobsthal matrix sequence
Int. J. Adv. Appl. Math. and Mech. 6(3 (2019 14 20 (ISSN: 2347-2529 Journal homepage: www.jaamm.com IJAAMM Internatonal Journal of Advances n Appled Mathematcs and Mechancs The bnomal transforms of the
More informationLecture 10: May 6, 2013
TTIC/CMSC 31150 Mathematcal Toolkt Sprng 013 Madhur Tulsan Lecture 10: May 6, 013 Scrbe: Wenje Luo In today s lecture, we manly talked about random walk on graphs and ntroduce the concept of graph expander,
More informationIntroduction to Algorithms
Introducton to Algorthms 6.046J/8.40J Lecture 7 Prof. Potr Indyk Data Structures Role of data structures: Encapsulate data Support certan operatons (e.g., INSERT, DELETE, SEARCH) Our focus: effcency of
More informationIntroductory Cardinality Theory Alan Kaylor Cline
Introductory Cardnalty Theory lan Kaylor Clne lthough by name the theory of set cardnalty may seem to be an offshoot of combnatorcs, the central nterest s actually nfnte sets. Combnatorcs deals wth fnte
More informationQuadratic Forms. Quadratic Forms
Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte
More informationLearning Theory: Lecture Notes
Learnng Theory: Lecture Notes Lecturer: Kamalka Chaudhur Scrbe: Qush Wang October 27, 2012 1 The Agnostc PAC Model Recall that one of the constrants of the PAC model s that the data dstrbuton has to be
More informationChemical Reaction Engineering
Lecture 20 hemcl Recton Engneerng (RE) s the feld tht studes the rtes nd mechnsms of chemcl rectons nd the desgn of the rectors n whch they tke plce. Lst Lecture Energy Blnce Fundmentls F 0 E 0 F E Q W
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
More informationZero-Sum Magic Graphs and Their Null Sets
Zero-Sum Mgic Grphs nd Their Null Sets Ebrhim Slehi Deprtment of Mthemticl Sciences University of Nevd Ls Vegs Ls Vegs, NV 89154-4020. ebrhim.slehi@unlv.edu Abstrct For ny h N, grph G = (V, E) is sid to
More informationProperties of the Riemann Integral
Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University Februry 15, 2018 Outline 1 Some Infimum nd Supremum Properties 2
More informationLecture 1: Introduction to integration theory and bounded variation
Lecture 1: Introduction to integrtion theory nd bounded vrition Wht is this course bout? Integrtion theory. The first question you might hve is why there is nything you need to lern bout integrtion. You
More informationN 0 completions on partial matrices
N 0 completions on prtil mtrices C. Jordán C. Mendes Arújo Jun R. Torregros Instituto de Mtemátic Multidisciplinr / Centro de Mtemátic Universidd Politécnic de Vlenci / Universidde do Minho Cmino de Ver
More informationIntroduction To Matrices MCV 4UI Assignment #1
Introduction To Mtrices MCV UI Assignment # INTRODUCTION: A mtrix plurl: mtrices) is rectngulr rry of numbers rrnged in rows nd columns Exmples: ) b) c) [ ] d) Ech number ppering in the rry is sid to be
More informationQuiz: Experimental Physics Lab-I
Mxmum Mrks: 18 Totl tme llowed: 35 mn Quz: Expermentl Physcs Lb-I Nme: Roll no: Attempt ll questons. 1. In n experment, bll of mss 100 g s dropped from heght of 65 cm nto the snd contner, the mpct s clled
More information4.4 Areas, Integrals and Antiderivatives
. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order
More informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More informationName: SID: Discussion Session:
Nme: SID: Dscusson Sesson: hemcl Engneerng hermodynmcs -- Fll 008 uesdy, Octoer, 008 Merm I - 70 mnutes 00 onts otl losed Book nd Notes (5 ponts). onsder n del gs wth constnt het cpctes. Indcte whether
More informationInfinite Geometric Series
Infinite Geometric Series Finite Geometric Series ( finite SUM) Let 0 < r < 1, nd let n be positive integer. Consider the finite sum It turns out there is simple lgebric expression tht is equivlent to
More informationarxiv:math/ v2 [math.ho] 16 Dec 2003
rxiv:mth/0312293v2 [mth.ho] 16 Dec 2003 Clssicl Lebesgue Integrtion Theorems for the Riemnn Integrl Josh Isrlowitz 244 Ridge Rd. Rutherford, NJ 07070 jbi2@njit.edu Februry 1, 2008 Abstrct In this pper,
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
More informationarxiv: v1 [math.co] 1 Mar 2014
Unon-ntersectng set systems Gyula O.H. Katona and Dánel T. Nagy March 4, 014 arxv:1403.0088v1 [math.co] 1 Mar 014 Abstract Three ntersecton theorems are proved. Frst, we determne the sze of the largest
More information