Folding of Regular CW-Complexes
|
|
- Elwin Warren
- 5 years ago
- Views:
Transcription
1 Ald Mathmatcal Scncs, Vol. 6,, no. 83, Foldng of Rgular CW-Comlxs E. M. El-Kholy and S N. Daoud,3. Dartmnt of Mathmatcs, Faculty of Scnc Tanta Unvrsty,Tanta,Egyt. Dartmnt of Mathmatcs, Faculty of Scnc, El-Mnufya Unvrsty Shbn El-Kom, Egyt 3. Dartmnt of Ald Mathmatcs, Faculty of Ald Scnc Tabah Unvrsty, Al-Madnah, Saud Araba sa_na_daoud@yahoo.com Abstract In ths ar w obtand som rlatons btwn th rgular CW-comlx K and ts mag such as homologcal grous, Btt numbrs, Eulr charactrstc and th gnus undr a cllular foldng, a nat cllular foldng, th comoston of two cllular foldngs and fnally th comoston of a cllular foldng and a nat cllular foldng.. Introducton A cllular foldng s a foldng dfnd on rgular CW-comlxs frst dfnd by E. El-Kholy and. Al-Khurasan, [], and varous rorts of ths ty of foldng ar also studd by thm. By a cllular foldng of rgular CWcomlxs, t s mant a cllular ma f : K L whch mas -clls of K to - clls of L and such that f, for ach -cll, s a homomorhsm onto ts mag. Th st of rgular CW-comlxs togthr wth cllular foldngs form a catgory dnotd by C ( K, L. If f C ( K, L ; thn x K s sad to b a sngularty of f ff f s not a local homomorhsm at x. Th st of all sngularts of f s dnotd by f. Ths st corrsonds to th folds of th ma. It s notcd that for a cllular foldng f, th st f of sngularts of f s a ror subst of th unon of clls of dmnsons n. Thus whn w consdr any f C ( K, L,
2 438 E. M. El-Kholy and S N. Daoud whr K and L ar connctd rgular CW-comlxs of dmnson, th st f wll conssts of -clls, and -clls, ach -cll (vrtx has an vn valncy, [, 3,6]. Of cours, f nd not b connctd. Now a nat cllular n n foldng f : K L s a cllular foldng such that L L conssts of a sngl n- cll, Int L, [4,5]. Th st of rgular CW-comlxs togthr wth nat cllular foldng from a catgory whch s dnotd by N C ( K, L. Ths catgory s a subcatgory of th catgory C ( K, L. Agan f f th st of sngularts of f and ths st corrsonds to th folds of th ma. From now w man by a comlx a rgular CW-comlx.. Man rsults Th followng thorm gvs som rlatons btwn a comlx and ts mag undr a cllular foldng. Thorm.: Lt K and L comlxs of th sam dmnsons n. If ϕ C ( K, L such that ϕ ( K = L K. Thn ( ( K ( L Krϕ, ( ( K = ( L + r k (Kr ϕ, ( χ ( K = χ ( L + ( r k (Kr ϕ, = g ( K = g ( L ( r k ( Kr ϕ (v and = ( = ( ( ( Kr = g K g L r k ϕ comlxs, whr ϕ : ( K ( L s th nducd homomorhsm. n cas of orntabl comlxs,, n cas of non-orntabl Proof: ( Consdr th nducd homomorhsm ϕ : ( K ( L, thr s a short xact squnc ϕ K ϕ Kr ϕ ( Im,
3 Foldng of rgular CW-comlxs 439 whr s th nducd homomorhsm by th ncluson. Snc ϕ s surjctv w hav Im ϕ ( L and ϕ ([ ] z = ϕ ([ z ] for all z n ( K, hnc th abov squnc wll b tak th form ϕ Kr ϕ ( K ( L. Ths squnc can b slt by th homomorhsm h: ( L ( K such that h( a = ϕ ( a for all a ( L and hnc w hav th rsult. ( From ( w hav ( K ( L Krϕ. Thus rk( ( K = rk( ( L Kr ϕ = rk( ( L + rk(kr ϕ. Thrfor ( K = ( L + rk (Kr ϕ ( Snc ( K = ( L + rk (Kr ϕ for =,,,..., n. Thn ( K = ( L + rk (Kr ϕ ( K = ( L + r k (Kr ϕ M M M n ( K = n ( L + r k (Kr ϕn Thus w hav + + K = + + L + rk φ = χ ( K = χ( L + ( rk (Kr ϕ. = (... ( ( (... ( ( ( (Kr Thrfor (v Snc g = ( x n cas of orntabl comlxs. Thn χ = g. Thus g ( K = g ( L + ( r k (Kr ϕ and hnc = g ( K = g ( L ( r k (Kr ϕ. = In cas of non orantabl comlxs, g = χ.., χ = g Thus g ( x = g ( L + ( r k ( Kr ϕ = Thn g ( K = g ( L ( r k (Kr ϕ. Examl. = (a Consdr th cllular foldng f of a comlx K such that K s a shr nto tslf wth cllular subdvson shown n Fg. (. Th mag of ths ma s a comlx L such that L s a dsc.
4 44 E. M. El-Kholy and S N. Daoud f K : Z : O : Z Fg. ( Z O O L Kr f Kr f Kr f It s asy to s that th condtons of thorm ( ar satsfd. (b Consdr th cllular foldng g of a comlx K such that K s a tours nto tslf wth cllular subdvson shown n Fg. (. Th mag f ths ma s a comlx L such that L s a cylndr K 7 6 g 4 L 6 5 Z : Z : Z Z : Z Kr g Fg.( A gan th condtons of thorm (. ar satsfd. Th followng thorm gvs som rlatons btwn a comlx and ts mag undr a nat cllular foldng.
5 Foldng of rgular CW-comlxs 44 ( ( Thorm.3: Lt K, L b comlxs of th sam dmnson n. If ϕ NC( K, L such that ϕ ( K = L K. Thn K ( Kr ϕ, ( K = rk (Kr ϕ ( χ ( K = + ( rk (Kr ϕ = (v g K = rk ϕ = and ( ( ( K r =, ncas of orntabl comlxs, g ( K = ( rk (Kr ϕ, ncas of non-orntabl comlxs, whr ϕ : ( K ( L s th nducd homomorhsm. Proof: ( Consdr th nducd homomorhsm ϕ : ( K ( L, thr s a short xact squnc ϕ Kr ϕ ( K Imϕ whr s th nducd homomorhsm by th ncluson. Snc ϕ s surjctv, w hav Im ϕ ( L, but ( L for a nat cllular foldng, hnc th abov squnc wll tak th form Kr ϕ ( K and th roof of ( wll follow thn from th xactnss of th squnc. ( From ( w hav ( K Kr ϕ, Thus rk ( ( K = rk (Kr ϕ, and hnc ( K = rk (Kr ϕ ( Snc ( K = rk (Kr ϕ, =,,..., n Thn ( K = rk (Kr ϕ ( K = rk (Kr ϕ M n ( K = rk (Kr ϕ n
6 44 E. M. El-Kholy and S N. Daoud Thus ( ( ( K = ( K + ( rk ( Kr ϕ, and hnc χ ( K = + ( rk (K r ϕ =. = (v g = ( χ for orntabl comlxs.., χ = g. Thus w hav g ( K = + ( rk (Kr ϕ = Thn g ( K = ( rk (Kr ϕ, and = g = χ for non-orntabl comlxs..., x = g Thus w hav g ( K = ( rk ( Kr ϕ. = Examl.4 (a Lt K b a comlx such that K s th rojctv lan and th ma f : K L s a nat cllular foldng such that f ( K homomorhc to a dsk. a a K Fg. (3 It s asy to chck that th condtons of thorm ( ar satsfd. (b Consdr a comlx K such that K = T # T, th doubl torus, wth cllular subdvson consstng of tn -clls, twnty -clls and ght -clls. S Fg. (4. Lt f : K K b a nat cllular foldng dfnd by: o o o o o o o f ( 6,..., = ( 4, 5, 3,, and th omttd -clls wll b mad nto tslf. f (, j, k, l, m = (,, 3, 4, 5, whr =,, 3, 4, j =, 9, 5, 6, k = 3, 8, 7, 8, l = 4, 7, 9,, m = 5, 6,, and f ( =, =,..., 8.
7 Foldng of rgular CW-comlxs f 4 7 K f ( K = L Fg. (4 Th mag f ( K = L, whr L s a dsc wth cllular subdvson consstng of fv -clls, fv -clls and a sngl -cll. Agan th condtons of thorm (.3 ar satsfd. Th followng thorm gvs som rlatons btwn a comlx and ts mag undr th comoston of two cllular foldngs. Thorm.5: Lt K, L and M b comlxs of th sam dmnson n. If f C ( K, L such that f ( K = L K and g C ( L, M such that g ( L = M L. Thn ( g o f C ( M, L and th followng condtons ar satsfd: ( ( K ( M Kr( g o f ( ( K = ( M + rk (Kr( g o f ( χ ( K = χ ( M + ( rk ( Kr ( g o f = g ( K = g ( M ( rk (Kr( g f (v o n cas of orntabl = comlxs and, g ( K = g ( M ( rk (Kr( g o f, n cas of nonorntabl comlxs; = whr ( g of = g o f : ( K ( M s th nducd homomorhsm.
8 444 E. M. El-Kholy and S N. Daoud Examl.6: Consdr th cllular foldng f of a comlx K such that K s a tours nto tslf wth cllular subdvson shown n Fg. (5. Th mag of ths ma s a comlx L such that L s a cylndr and nto tslf. S Fg. ( K f 7 6 L 6 4 g M 4 Fg. (5 It s asy to chck that th condtons of thorm (.5 ar satsfd. Th followng thorm gvs som rlatons btwn a comlx and ts mag undr comoston of a cllular foldng and a nat cllular foldng. ( ( Thorm.7: Lt K, L and M b comlxs of th sam dmnson n. If f C( K, L such that f ( K = L K, g NC( L, M such that g ( L = M L. Thn g o f N C ( K, M and th followng condtons ar satsfd: K g f ( Kr( o, K = r k g o f ( (Kr( ( x ( K = + ( rk (Kr( g o f = (v g ( K = ( rk (Kr( g o f n cas of orntabl comlxs, and whr = ( = ( (Kr ( = = g K rk g f o n cas of non-orntabl comlxs, ( g of g o f : ( K ( M s th nduc homomorhsm.
9 Foldng of rgular CW-comlxs 445 Examl.8 : Consdr th comlxs K, L and M such that K, L and M ar tours, cylndr and dsc rsctvly wth cllular subdvson shown n Fg. (6 and lt f C ( K, L, g N C ( L, M f g K L M Fg. (6 6 It s asy to chck th condtons of thorm (.7 ar satsfd. Rfrncs [] E. M. El-Kholy and Al-Kurasan,. A., Foldng of CW-comlxs, J. Inst. Math & Com. Sc. (Math. Sr., Inda Vol. 4 No.,. 4-48, (99. []. R. Farran, E. El. Kholy and S. A. Robrtson, Foldng a surfac to a olygon, Gomtra Ddcata V. 33, , (996. [3] S. A. Robrtson and E. El-Kholy, Toologcal Foldng, commum, Fac. Sc. Unv. Ank. Srs A V. 35,. -7, (986. [4] E. M El-Kholy. and Shahn, R. M., Cllular foldng, J. Inst. Math. & Com. Sc. (Math. Sr. Vol., No. 3,. 77-8, Inda (998.
10 446 E. M. El-Kholy and S N. Daoud [5] E. M El-Kholy., S.R Lashn and S.N Daoud: Equ- Gauss curvatur foldng, Proc. Indan Acad. Sc.(Math. Sc. Vol.7, No3,.93-3, Inda August (7. [6] A.atchr, Algbrac Toology, Cambrdg Unvrsty rss, London (. Rcvd: March,
ON THE COMPLEXITY OF K-STEP AND K-HOP DOMINATING SETS IN GRAPHS
MATEMATICA MONTISNIRI Vol XL (2017) MATEMATICS ON TE COMPLEXITY OF K-STEP AN K-OP OMINATIN SETS IN RAPS M FARAI JALALVAN AN N JAFARI RA partmnt of Mathmatcs Shahrood Unrsty of Tchnology Shahrood Iran Emals:
More informationLucas Test is based on Euler s theorem which states that if n is any integer and a is coprime to n, then a φ(n) 1modn.
Modul 10 Addtonal Topcs 10.1 Lctur 1 Prambl: Dtrmnng whthr a gvn ntgr s prm or compost s known as prmalty tstng. Thr ar prmalty tsts whch mrly tll us whthr a gvn ntgr s prm or not, wthout gvng us th factors
More informationHardy-Littlewood Conjecture and Exceptional real Zero. JinHua Fei. ChangLing Company of Electronic Technology Baoji Shannxi P.R.
Hardy-Littlwood Conjctur and Excptional ral Zro JinHua Fi ChangLing Company of Elctronic Tchnology Baoji Shannxi P.R.China E-mail: fijinhuayoujian@msn.com Abstract. In this papr, w assum that Hardy-Littlwood
More informationWeek 3: Connected Subgraphs
Wk 3: Connctd Subgraphs Sptmbr 19, 2016 1 Connctd Graphs Path, Distanc: A path from a vrtx x to a vrtx y in a graph G is rfrrd to an xy-path. Lt X, Y V (G). An (X, Y )-path is an xy-path with x X and y
More informationGroup Codes Define Over Dihedral Groups of Small Order
Malaysan Journal of Mathmatcal Scncs 7(S): 0- (0) Spcal Issu: Th rd Intrnatonal Confrnc on Cryptology & Computr Scurty 0 (CRYPTOLOGY0) MALAYSIA JOURAL OF MATHEMATICAL SCIECES Journal hompag: http://nspm.upm.du.my/ournal
More informationThe Hyperelastic material is examined in this section.
4. Hyprlastcty h Hyprlastc matral s xad n ths scton. 4..1 Consttutv Equatons h rat of chang of ntrnal nrgy W pr unt rfrnc volum s gvn by th strss powr, whch can b xprssd n a numbr of dffrnt ways (s 3.7.6):
More informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #20 11/14/2013
18.782 Introduction to Arithmtic Gomtry Fall 2013 Lctur #20 11/14/2013 20.1 Dgr thorm for morphisms of curvs Lt us rstat th thorm givn at th nd of th last lctur, which w will now prov. Thorm 20.1. Lt φ:
More informationGrand Canonical Ensemble
Th nsmbl of systms mmrsd n a partcl-hat rsrvor at constant tmpratur T, prssur P, and chmcal potntal. Consdr an nsmbl of M dntcal systms (M =,, 3,...M).. Thy ar mutually sharng th total numbr of partcls
More informationCombinatorial Networks Week 1, March 11-12
1 Nots on March 11 Combinatorial Ntwors W 1, March 11-1 11 Th Pigonhol Principl Th Pigonhol Principl If n objcts ar placd in hols, whr n >, thr xists a box with mor than on objcts 11 Thorm Givn a simpl
More informationThe Equitable Dominating Graph
Intrnational Journal of Enginring Rsarch and Tchnology. ISSN 0974-3154 Volum 8, Numbr 1 (015), pp. 35-4 Intrnational Rsarch Publication Hous http://www.irphous.com Th Equitabl Dominating Graph P.N. Vinay
More informationSection 6.1. Question: 2. Let H be a subgroup of a group G. Then H operates on G by left multiplication. Describe the orbits for this operation.
MAT 444 H Barclo Spring 004 Homwork 6 Solutions Sction 6 Lt H b a subgroup of a group G Thn H oprats on G by lft multiplication Dscrib th orbits for this opration Th orbits of G ar th right costs of H
More informationA NEW GENERALISATION OF SAM-SOLAI S MULTIVARIATE ADDITIVE GAMMA DISTRIBUTION*
A NEW GENERALISATION OF SAM-SOLAI S MULTIVARIATE ADDITIVE GAMMA DISTRIBUTION* Dr. G.S. Davd Sam Jayakumar, Assstant Profssor, Jamal Insttut of Managmnt, Jamal Mohamd Collg, Truchraall 620 020, South Inda,
More information4D SIMPLICIAL QUANTUM GRAVITY
T.YUKAWA and S.HORATA Soknda/KEK D SIMPLICIAL QUATUM GRAITY Plan of th talk Rvw of th D slcal quantu gravty Rvw of nurcal thods urcal rsult and dscusson Whr dos th slcal quantu gravty stand? In short dstanc
More informationA Note on Estimability in Linear Models
Intrnatonal Journal of Statstcs and Applcatons 2014, 4(4): 212-216 DOI: 10.5923/j.statstcs.20140404.06 A Not on Estmablty n Lnar Modls S. O. Adymo 1,*, F. N. Nwob 2 1 Dpartmnt of Mathmatcs and Statstcs,
More informationAn Application of Hardy-Littlewood Conjecture. JinHua Fei. ChangLing Company of Electronic Technology Baoji Shannxi P.R.China
An Application of Hardy-Littlwood Conjctur JinHua Fi ChangLing Company of Elctronic Tchnology Baoji Shannxi P.R.China E-mail: fijinhuayoujian@msn.com Abstract. In this papr, w assum that wakr Hardy-Littlwood
More informationcycle that does not cross any edges (including its own), then it has at least
W prov th following thorm: Thorm If a K n is drawn in th plan in such a way that it has a hamiltonian cycl that dos not cross any dgs (including its own, thn it has at last n ( 4 48 π + O(n crossings Th
More informationSCHUR S THEOREM REU SUMMER 2005
SCHUR S THEOREM REU SUMMER 2005 1. Combinatorial aroach Prhas th first rsult in th subjct blongs to I. Schur and dats back to 1916. On of his motivation was to study th local vrsion of th famous quation
More informationMath 656 March 10, 2011 Midterm Examination Solutions
Math 656 March 0, 0 Mdtrm Eamnaton Soltons (4pts Dr th prsson for snh (arcsnh sng th dfnton of snh w n trms of ponntals, and s t to fnd all als of snh (. Plot ths als as ponts n th compl plan. Mak sr or
More informationON RIGHT(LEFT) DUO PO-SEMIGROUPS. S. K. Lee and K. Y. Park
Kangwon-Kyungki Math. Jour. 11 (2003), No. 2, pp. 147 153 ON RIGHT(LEFT) DUO PO-SEMIGROUPS S. K. L and K. Y. Park Abstract. W invstigat som proprtis on right(rsp. lft) duo po-smigroups. 1. Introduction
More informationInternational Journal of Scientific & Engineering Research, Volume 6, Issue 7, July ISSN
Intrnational Journal of Scintific & Enginring Rsarch, Volum 6, Issu 7, July-25 64 ISSN 2229-558 HARATERISTIS OF EDGE UTSET MATRIX OF PETERSON GRAPH WITH ALGEBRAI GRAPH THEORY Dr. G. Nirmala M. Murugan
More informationIndependent Domination in Line Graphs
Itratoal Joural of Sctfc & Egrg Rsarch Volum 3 Issu 6 Ju-1 1 ISSN 9-5518 Iddt Domato L Grahs M H Muddbhal ad D Basavarajaa Abstract - For ay grah G th l grah L G H s th trscto grah Thus th vrtcs of LG
More informationOn Tiling for Some Types of Manifolds. and their Folding
Appled Mathematcal Scences, Vol. 3, 009, no. 6, 75-84 On Tlng for Some Types of Manfolds and ther Foldng H. Rafat Mathematcs Department, Faculty of Scence Tanta Unversty, Tanta Egypt hshamrafat005@yahoo.com
More informationHeisenberg Model. Sayed Mohammad Mahdi Sadrnezhaad. Supervisor: Prof. Abdollah Langari
snbrg Modl Sad Mohammad Mahd Sadrnhaad Survsor: Prof. bdollah Langar bstract: n ths rsarch w tr to calculat analtcall gnvalus and gnvctors of fnt chan wth ½-sn artcls snbrg modl. W drov gnfuctons for closd
More informationMatched Quick Switching Variable Sampling System with Quick Switching Attribute Sampling System
Natur and Sn 9;7( g v, t al, Samlng Systm Mathd Quk Swthng Varabl Samlng Systm wth Quk Swthng Attrbut Samlng Systm Srramahandran G.V, Palanvl.M Dartmnt of Mathmats, Dr.Mahalngam Collg of Engnrng and Thnology,
More informationSOME PARAMETERS ON EQUITABLE COLORING OF PRISM AND CIRCULANT GRAPH.
SOME PARAMETERS ON EQUITABLE COLORING OF PRISM AND CIRCULANT GRAPH. K VASUDEVAN, K. SWATHY AND K. MANIKANDAN 1 Dpartmnt of Mathmatics, Prsidncy Collg, Chnnai-05, India. E-Mail:vasu k dvan@yahoo.com. 2,
More informationFigure 1: Closed surface, surface with boundary, or not a surface?
QUESTION 1 (10 marks) Two o th topological spacs shown in Figur 1 ar closd suracs, two ar suracs with boundary, and two ar not suracs. Dtrmin which is which. You ar not rquird to justiy your answr, but,
More informationDerangements and Applications
2 3 47 6 23 Journal of Intgr Squncs, Vol. 6 (2003), Articl 03..2 Drangmnts and Applications Mhdi Hassani Dpartmnt of Mathmatics Institut for Advancd Studis in Basic Scincs Zanjan, Iran mhassani@iasbs.ac.ir
More informationRecall that by Theorems 10.3 and 10.4 together provide us the estimate o(n2 ), S(q) q 9, q=1
Chaptr 11 Th singular sris Rcall that by Thorms 10 and 104 togthr provid us th stimat 9 4 n 2 111 Rn = SnΓ 2 + on2, whr th singular sris Sn was dfind in Chaptr 10 as Sn = q=1 Sq q 9, with Sq = 1 a q gcda,q=1
More informationPhysics 256: Lecture 2. Physics
Physcs 56: Lctur Intro to Quantum Physcs Agnda for Today Complx Numbrs Intrfrnc of lght Intrfrnc Two slt ntrfrnc Dffracton Sngl slt dffracton Physcs 01: Lctur 1, Pg 1 Constructv Intrfrnc Ths wll occur
More informationPropositional Logic. Combinatorial Problem Solving (CPS) Albert Oliveras Enric Rodríguez-Carbonell. May 17, 2018
Propositional Logic Combinatorial Problm Solving (CPS) Albrt Olivras Enric Rodríguz-Carbonll May 17, 2018 Ovrviw of th sssion Dfinition of Propositional Logic Gnral Concpts in Logic Rduction to SAT CNFs
More informationDISTRIBUTION OF DIFFERENCE BETWEEN INVERSES OF CONSECUTIVE INTEGERS MODULO P
DISTRIBUTION OF DIFFERENCE BETWEEN INVERSES OF CONSECUTIVE INTEGERS MODULO P Tsz Ho Chan Dartmnt of Mathmatics, Cas Wstrn Rsrv Univrsity, Clvland, OH 4406, USA txc50@cwru.du Rcivd: /9/03, Rvisd: /9/04,
More informationINCOMPLETE KLOOSTERMAN SUMS AND MULTIPLICATIVE INVERSES IN SHORT INTERVALS. xy 1 (mod p), (x, y) I (j)
INCOMPLETE KLOOSTERMAN SUMS AND MULTIPLICATIVE INVERSES IN SHORT INTERVALS T D BROWNING AND A HAYNES Abstract W invstigat th solubility of th congrunc xy (mod ), whr is a rim and x, y ar rstrictd to li
More informationCOMPLEX NUMBER PAIRWISE COMPARISON AND COMPLEX NUMBER AHP
ISAHP 00, Bal, Indonsa, August -9, 00 COMPLEX NUMBER PAIRWISE COMPARISON AND COMPLEX NUMBER AHP Chkako MIYAKE, Kkch OHSAWA, Masahro KITO, and Masaak SHINOHARA Dpartmnt of Mathmatcal Informaton Engnrng
More informationSeptember 27, Introduction to Ordinary Differential Equations. ME 501A Seminar in Engineering Analysis Page 1. Outline
Introucton to Ornar Dffrntal Equatons Sptmbr 7, 7 Introucton to Ornar Dffrntal Equatons Larr artto Mchancal Engnrng AB Smnar n Engnrng Analss Sptmbr 7, 7 Outln Rvw numrcal solutons Bascs of ffrntal quatons
More informationte Finance (4th Edition), July 2017.
Appndx Chaptr. Tchncal Background Gnral Mathmatcal and Statstcal Background Fndng a bas: 3 2 = 9 3 = 9 1 /2 x a = b x = b 1/a A powr of 1 / 2 s also quvalnt to th squar root opraton. Fndng an xponnt: 3
More informationAnnounce. ECE 2026 Summer LECTURE OBJECTIVES READING. LECTURE #3 Complex View of Sinusoids May 21, Complex Number Review
ECE 06 Summr 018 Announc HW1 du at bginning of your rcitation tomorrow Look at HW bfor rcitation Lab 1 is Thursday: Com prpard! Offic hours hav bn postd: LECTURE #3 Complx Viw of Sinusoids May 1, 018 READIG
More informationON EISENSTEIN-DUMAS AND GENERALIZED SCHÖNEMANN POLYNOMIALS
ON EISENSTEIN-DUMAS AND GENERALIZED SCHÖNEMANN POLYNOMIALS Anuj Bshno and Sudsh K. Khanduja Dpartmnt of Mathmatcs, Panjab Unvrsty, Chandgarh-160014, Inda. E-mal: anuj.bshn@gmal.com, skhand@pu.ac.n ABSTRACT.
More informationJEE-2017 : Advanced Paper 2 Answers and Explanations
DE 9 JEE-07 : Advancd Papr Answrs and Explanatons Physcs hmstry Mathmatcs 0 A, B, 9 A 8 B, 7 B 6 B, D B 0 D 9, D 8 D 7 A, B, D A 0 A,, D 9 8 * A A, B A B, D 0 B 9 A, D 5 D A, B A,B,,D A 50 A, 6 5 A D B
More informationBINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES
BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES DONALD M. DAVIS Abstract. If p is a prim (implicit in notation and n a positiv intgr, lt ν(n dnot th xponnt of p in n, and U(n n/p ν(n, th unit
More informationReview - Probabilistic Classification
Mmoral Unvrsty of wfoundland Pattrn Rcognton Lctur 8 May 5, 6 http://www.ngr.mun.ca/~charlsr Offc Hours: Tusdays Thursdays 8:3-9:3 PM E- (untl furthr notc) Gvn lablld sampls { ɛc,,,..., } {. Estmat Rvw
More informationarxiv: v1 [math.pr] 28 Jan 2019
CRAMÉR-TYPE MODERATE DEVIATION OF NORMAL APPROXIMATION FOR EXCHANGEABLE PAIRS arxv:190109526v1 [mathpr] 28 Jan 2019 ZHUO-SONG ZHANG Abstract In Stn s mthod, an xchangabl par approach s commonly usd to
More informationDepartment of Mathematics and Statistics Indian Institute of Technology Kanpur MSO202A/MSO202 Assignment 3 Solutions Introduction To Complex Analysis
Dpartmt of Mathmatcs ad Statstcs Ida Isttut of Tchology Kapur MSOA/MSO Assgmt 3 Solutos Itroducto To omplx Aalyss Th problms markd (T) d a xplct dscusso th tutoral class. Othr problms ar for hacd practc..
More informationAddition of angular momentum
Addition of angular momntum April, 0 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat th
More informationEAcos θ, where θ is the angle between the electric field and
8.4. Modl: Th lctric flux flows out of a closd surfac around a rgion of spac containing a nt positiv charg and into a closd surfac surrounding a nt ngativ charg. Visualiz: Plas rfr to Figur EX8.4. Lt A
More informationAnalyzing Frequencies
Frquncy (# ndvduals) Frquncy (# ndvduals) /3/16 H o : No dffrnc n obsrvd sz frquncs and that prdctd by growth modl How would you analyz ths data? 15 Obsrvd Numbr 15 Expctd Numbr from growth modl 1 1 5
More informationGEOMETRICAL PHENOMENA IN THE PHYSICS OF SUBATOMIC PARTICLES. Eduard N. Klenov* Rostov-on-Don, Russia
GEOMETRICAL PHENOMENA IN THE PHYSICS OF SUBATOMIC PARTICLES Eduard N. Klnov* Rostov-on-Don, Russia Th articl considrs phnomnal gomtry figurs bing th carrirs of valu spctra for th pairs of th rmaining additiv
More informationΕρωτήσεις και ασκησεις Κεφ. 10 (για μόρια) ΠΑΡΑΔΟΣΗ 29/11/2016. (d)
Ερωτήσεις και ασκησεις Κεφ 0 (για μόρια ΠΑΡΑΔΟΣΗ 9//06 Th coffcnt A of th van r Waals ntracton s: (a A r r / ( r r ( (c a a a a A r r / ( r r ( a a a a A r r / ( r r a a a a A r r / ( r r 4 a a a a 0 Th
More information10. EXTENDING TRACTABILITY
Coping with NP-compltnss 0. EXTENDING TRACTABILITY ining small vrtx covrs solving NP-har problms on trs circular arc covrings vrtx covr in bipartit graphs Q. Suppos I n to solv an NP-complt problm. What
More information10/7/14. Mixture Models. Comp 135 Introduction to Machine Learning and Data Mining. Maximum likelihood estimation. Mixture of Normals in 1D
Comp 35 Introducton to Machn Larnng and Data Mnng Fall 204 rofssor: Ron Khardon Mxtur Modls Motvatd by soft k-mans w dvlopd a gnratv modl for clustrng. Assum thr ar k clustrs Clustrs ar not rqurd to hav
More information10. The Discrete-Time Fourier Transform (DTFT)
Th Discrt-Tim Fourir Transform (DTFT Dfinition of th discrt-tim Fourir transform Th Fourir rprsntation of signals plays an important rol in both continuous and discrt signal procssing In this sction w
More informationVISUALIZATION OF DIFFERENTIAL GEOMETRY UDC 514.7(045) : : Eberhard Malkowsky 1, Vesna Veličković 2
FACTA UNIVERSITATIS Srs: Mchancs, Automatc Control Robotcs Vol.3, N o, 00, pp. 7-33 VISUALIZATION OF DIFFERENTIAL GEOMETRY UDC 54.7(045)54.75.6:59.688:59.673 Ebrhard Malkowsky, Vsna Vlčkovć Dpartmnt of
More informationCOHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim.
MTH rviw part b Lucian Mitroiu Th LOG and EXP functions Th ponntial function p : R, dfind as Proprtis: lim > lim p Eponntial function Y 8 6 - -8-6 - - X Th natural logarithm function ln in US- log: function
More informationConsider a system of 2 simultaneous first order linear equations
Soluon of sysms of frs ordr lnar quaons onsdr a sysm of smulanous frs ordr lnar quaons a b c d I has h alrna mar-vcor rprsnaon a b c d Or, n shorhand A, f A s alrady known from con W know ha h abov sysm
More informationZero Point Energy: Thermodynamic Equilibrium and Planck Radiation Law
Gaug Institut Journa Vo. No 4, Novmbr 005, Zro Point Enrgy: Thrmodynamic Equiibrium and Panck Radiation Law Novmbr, 005 vick@adnc.com Abstract: In a rcnt papr, w provd that Panck s radiation aw with zro
More informationA general N-dimensional vector consists of N values. They can be arranged as a column or a row and can be real or complex.
Lnr lgr Vctors gnrl -dmnsonl ctor conssts of lus h cn rrngd s column or row nd cn rl or compl Rcll -dmnsonl ctor cn rprsnt poston, loct, or cclrton Lt & k,, unt ctors long,, & rspctl nd lt k h th componnts
More informationJournal of Theoretical and Applied Information Technology 10 th January Vol. 47 No JATIT & LLS. All rights reserved.
Journal o Thortcal and Appld Inormaton Tchnology th January 3. Vol. 47 No. 5-3 JATIT & LLS. All rghts rsrvd. ISSN: 99-8645 www.att.org E-ISSN: 87-395 RESEARCH ON PROPERTIES OF E-PARTIAL DERIVATIVE OF LOGIC
More informationSpectral Synthesis in the Heisenberg Group
Intrnational Journal of Mathmatical Analysis Vol. 13, 19, no. 1, 1-5 HIKARI Ltd, www.m-hikari.com https://doi.org/1.1988/ijma.19.81179 Spctral Synthsis in th Hisnbrg Group Yitzhak Wit Dpartmnt of Mathmatics,
More informationTwo Types of Geometric-Arithmetic Indices of Nanotubes and Nanotori
MACH Concatons n Mathatcal and n Cotr Chstry MACH Con Math Cot Ch 9 0 5-74 IN 040-5 wo ys of Gotrc-Arthtc Indcs of Nanotbs and Nanotor ros Morad, oraya Baba-Rah Dartnt of Mathatcs, Faclty of cnc, Arak
More informationMutually Independent Hamiltonian Cycles of Pancake Networks
Mutually Indpndnt Hamiltonian Cycls of Pancak Ntworks Chng-Kuan Lin Dpartmnt of Mathmatics National Cntral Univrsity, Chung-Li, Taiwan 00, R O C discipl@ms0urlcomtw Hua-Min Huang Dpartmnt of Mathmatics
More informationUNIT 8 TWO-WAY ANOVA WITH m OBSERVATIONS PER CELL
UNIT 8 TWO-WAY ANOVA WITH OBSERVATIONS PER CELL Two-Way Anova wth Obsrvatons Pr Cll Structur 81 Introducton Obctvs 8 ANOVA Modl for Two-way Classfd Data wth Obsrvatons r Cll 83 Basc Assutons 84 Estaton
More informationLimiting value of higher Mahler measure
Limiting valu of highr Mahlr masur Arunabha Biswas a, Chris Monico a, a Dpartmnt of Mathmatics & Statistics, Txas Tch Univrsity, Lubbock, TX 7949, USA Abstract W considr th k-highr Mahlr masur m k P )
More informationStrongly Connected Components
Strongly Connctd Componnts Lt G = (V, E) b a dirctd graph Writ if thr is a path from to in G Writ if and is an quivalnc rlation: implis and implis s quivalnc classs ar calld th strongly connctd componnts
More informationElectrochemical Equilibrium Electromotive Force. Relation between chemical and electric driving forces
C465/865, 26-3, Lctur 7, 2 th Sp., 26 lctrochmcal qulbrum lctromotv Forc Rlaton btwn chmcal and lctrc drvng forcs lctrochmcal systm at constant T and p: consdr G Consdr lctrochmcal racton (nvolvng transfr
More information1 Minimum Cut Problem
CS 6 Lctur 6 Min Cut and argr s Algorithm Scribs: Png Hui How (05), Virginia Dat: May 4, 06 Minimum Cut Problm Today, w introduc th minimum cut problm. This problm has many motivations, on of which coms
More informationNote If the candidate believes that e x = 0 solves to x = 0 or gives an extra solution of x = 0, then withhold the final accuracy mark.
. (a) Eithr y = or ( 0, ) (b) Whn =, y = ( 0 + ) = 0 = 0 ( + ) = 0 ( )( ) = 0 Eithr = (for possibly abov) or = A 3. Not If th candidat blivs that = 0 solvs to = 0 or givs an tra solution of = 0, thn withhold
More informationSquare of Hamilton cycle in a random graph
Squar of Hamilton cycl in a random graph Andrzj Dudk Alan Friz Jun 28, 2016 Abstract W show that p = n is a sharp thrshold for th random graph G n,p to contain th squar of a Hamilton cycl. This improvs
More informationTotal Prime Graph. Abstract: We introduce a new type of labeling known as Total Prime Labeling. Graphs which admit a Total Prime labeling are
Itratoal Joural Of Computatoal Egrg Rsarch (crol.com) Vol. Issu. 5 Total Prm Graph M.Rav (a) Ramasubramaa 1, R.Kala 1 Dpt.of Mathmatcs, Sr Shakth Isttut of Egrg & Tchology, Combator 641 06. Dpt. of Mathmatcs,
More informationOptimal Ordering Policy in a Two-Level Supply Chain with Budget Constraint
Optmal Ordrng Polcy n a Two-Lvl Supply Chan wth Budgt Constrant Rasoul aj Alrza aj Babak aj ABSTRACT Ths papr consdrs a two- lvl supply chan whch consst of a vndor and svral rtalrs. Unsatsfd dmands n rtalrs
More informationnd the particular orthogonal trajectory from the family of orthogonal trajectories passing through point (0; 1).
Eamn EDO. Givn th family of curvs y + C nd th particular orthogonal trajctory from th family of orthogonal trajctoris passing through point (0; ). Solution: In th rst plac, lt us calculat th di rntial
More informationShortest Paths in Graphs. Paths in graphs. Shortest paths CS 445. Alon Efrat Slides courtesy of Erik Demaine and Carola Wenk
S 445 Shortst Paths n Graphs lon frat Sls courtsy of rk man an arola Wnk Paths n raphs onsr a raph G = (V, ) wth -wht functon w : R. Th wht of path p = v v v k s fn to xampl: k = w ( p) = w( v, v + ).
More informationThe second condition says that a node α of the tree has exactly n children if the arity of its label is n.
CS 6110 S14 Hanout 2 Proof of Conflunc 27 January 2014 In this supplmntary lctur w prov that th λ-calculus is conflunt. This is rsult is u to lonzo Church (1903 1995) an J. arkly Rossr (1907 1989) an is
More informationy = 2xe x + x 2 e x at (0, 3). solution: Since y is implicitly related to x we have to use implicit differentiation: 3 6y = 0 y = 1 2 x ln(b) ln(b)
4. y = y = + 5. Find th quation of th tangnt lin for th function y = ( + ) 3 whn = 0. solution: First not that whn = 0, y = (1 + 1) 3 = 8, so th lin gos through (0, 8) and thrfor its y-intrcpt is 8. y
More informationBasic Polyhedral theory
Basic Polyhdral thory Th st P = { A b} is calld a polyhdron. Lmma 1. Eithr th systm A = b, b 0, 0 has a solution or thr is a vctorπ such that π A 0, πb < 0 Thr cass, if solution in top row dos not ist
More informationDeift/Zhou Steepest descent, Part I
Lctur 9 Dift/Zhou Stpst dscnt, Part I W now focus on th cas of orthogonal polynomials for th wight w(x) = NV (x), V (x) = t x2 2 + x4 4. Sinc th wight dpnds on th paramtr N N w will writ π n,n, a n,n,
More informationOn spanning trees and cycles of multicolored point sets with few intersections
On spanning trs and cycls of multicolord point sts with fw intrsctions M. Kano, C. Mrino, and J. Urrutia April, 00 Abstract Lt P 1,..., P k b a collction of disjoint point sts in R in gnral position. W
More informationAlpha and beta decay equation practice
Alpha and bta dcay quation practic Introduction Alpha and bta particls may b rprsntd in quations in svral diffrnt ways. Diffrnt xam boards hav thir own prfrnc. For xampl: Alpha Bta α β alpha bta Dspit
More information4. (5a + b) 7 & x 1 = (3x 1)log 10 4 = log (M1) [4] d = 3 [4] T 2 = 5 + = 16 or or 16.
. 7 7 7... 7 7 (n )0 7 (M) 0(n ) 00 n (A) S ((7) 0(0)) (M) (7 00) 8897 (A). (5a b) 7 7... (5a)... (M) 7 5 5 (a b ) 5 5 a b (M)(A) So th cofficint is 75 (A) (C) [] S (7 7) (M) () 8897 (A) (C) [] 5. x.55
More informationAddition of angular momentum
Addition of angular momntum April, 07 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat
More information9.5 Complex variables
9.5 Cmpl varabls. Cnsdr th funtn u v f( ) whr ( ) ( ), f( ), fr ths funtn tw statmnts ar as fllws: Statmnt : f( ) satsf Cauh mann quatn at th rgn. Statmnt : f ( ) ds nt st Th rrt statmnt ar (A) nl (B)
More informationCHAPTER 33: PARTICLE PHYSICS
Collg Physcs Studnt s Manual Chaptr 33 CHAPTER 33: PARTICLE PHYSICS 33. THE FOUR BASIC FORCES 4. (a) Fnd th rato of th strngths of th wak and lctromagntc forcs undr ordnary crcumstancs. (b) What dos that
More informationMath 656 Midterm Examination March 27, 2015 Prof. Victor Matveev
Math 656 Mdtrm Examnatn March 7, 05 Prf. Vctr Matvv ) (4pts) Fnd all vals f n plar r artsan frm, and plt thm as pnts n th cmplx plan: (a) Snc n-th rt has xactly n vals, thr wll b xactly =6 vals, lyng n
More informationFrequency Response. Response of an LTI System to Eigenfunction
Frquncy Rsons Las m w Rvsd formal dfnons of lnary and m-nvaranc Found an gnfuncon for lnar m-nvaran sysms Found h frquncy rsons of a lnar sysm o gnfuncon nu Found h frquncy rsons for cascad, fdbac, dffrnc
More informationSearch sequence databases 3 10/25/2016
Sarch squnc databass 3 10/25/2016 Etrm valu distribution Ø Suppos X is a random variabl with probability dnsity function p(, w sampl a larg numbr S of indpndnt valus of X from this distribution for an
More informationBSc Engineering Sciences A. Y. 2017/18 Written exam of the course Mathematical Analysis 2 August 30, x n, ) n 2
BSc Enginring Scincs A. Y. 27/8 Writtn xam of th cours Mathmatical Analysis 2 August, 28. Givn th powr sris + n + n 2 x n, n n dtrmin its radius of convrgnc r, and study th convrgnc for x ±r. By th root
More informationON THE INTEGRAL INVARIANTS OF KINEMATICALLY GENERATED RULED SURFACES *
Iranan Journal of Scnc & Tchnology Transacton A ol 9 No A Prntd n Th Islamc Rpublc of Iran 5 Shraz Unvrsty ON TH INTGRAL INARIANTS OF KINMATICALLY GNRATD RULD SURFACS H B KARADAG AND S KLS Dpartmnt of
More informationLecture 3: Phasor notation, Transfer Functions. Context
EECS 5 Fall 4, ctur 3 ctur 3: Phasor notaton, Transfr Functons EECS 5 Fall 3, ctur 3 Contxt In th last lctur, w dscussd: how to convrt a lnar crcut nto a st of dffrntal quatons, How to convrt th st of
More informationSearching Linked Lists. Perfect Skip List. Building a Skip List. Skip List Analysis (1) Assume the list is sorted, but is stored in a linked list.
3 3 4 8 6 3 3 4 8 6 3 3 4 8 6 () (d) 3 Sarching Linkd Lists Sarching Linkd Lists Sarching Linkd Lists ssum th list is sortd, but is stord in a linkd list. an w us binary sarch? omparisons? Work? What if
More informationEpistemic Foundations of Game Theory. Lecture 1
Royal Nthrlands cadmy of rts and Scncs (KNW) Mastr Class mstrdam, Fbruary 8th, 2007 Epstmc Foundatons of Gam Thory Lctur Gacomo onanno (http://www.con.ucdavs.du/faculty/bonanno/) QUESTION: What stratgs
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Procssing Prof. Mark Fowlr Dtails of th ot St #19 Rading Assignmnt: Sct. 7.1.2, 7.1.3, & 7.2 of Proakis & Manolakis Dfinition of th So Givn signal data points x[n] for n = 0,, -1
More information(Upside-Down o Direct Rotation) β - Numbers
Amrican Journal of Mathmatics and Statistics 014, 4(): 58-64 DOI: 10593/jajms0140400 (Upsid-Down o Dirct Rotation) β - Numbrs Ammar Sddiq Mahmood 1, Shukriyah Sabir Ali,* 1 Dpartmnt of Mathmatics, Collg
More informationRandom Process Part 1
Random Procss Part A random procss t (, ζ is a signal or wavform in tim. t : tim ζ : outcom in th sampl spac Each tim w rapat th xprimnt, a nw wavform is gnratd. ( W will adopt t for short. Tim sampls
More informationOutlier-tolerant parameter estimation
Outlr-tolrant paramtr stmaton Baysan thods n physcs statstcs machn larnng and sgnal procssng (SS 003 Frdrch Fraundorfr fraunfr@cg.tu-graz.ac.at Computr Graphcs and Vson Graz Unvrsty of Tchnology Outln
More informationExternal Equivalent. EE 521 Analysis of Power Systems. Chen-Ching Liu, Boeing Distinguished Professor Washington State University
xtrnal quvalnt 5 Analyss of Powr Systms Chn-Chng Lu, ong Dstngushd Profssor Washngton Stat Unvrsty XTRNAL UALNT ach powr systm (ara) s part of an ntrconnctd systm. Montorng dvcs ar nstalld and data ar
More informationFirst derivative analysis
Robrto s Nots on Dirntial Calculus Chaptr 8: Graphical analysis Sction First drivativ analysis What you nd to know alrady: How to us drivativs to idntiy th critical valus o a unction and its trm points
More informationMor Tutorial at www.dumblittldoctor.com Work th problms without a calculator, but us a calculator to chck rsults. And try diffrntiating your answrs in part III as a usful chck. I. Applications of Intgration
More informationECE602 Exam 1 April 5, You must show ALL of your work for full credit.
ECE62 Exam April 5, 27 Nam: Solution Scor: / This xam is closd-book. You must show ALL of your work for full crdit. Plas rad th qustions carfully. Plas chck your answrs carfully. Calculators may NOT b
More informationBasic Electrical Engineering for Welding [ ] --- Introduction ---
Basc Elctrcal Engnrng for Wldng [] --- Introducton --- akayosh OHJI Profssor Ertus, Osaka Unrsty Dr. of Engnrng VIUAL WELD CO.,LD t-ohj@alc.co.jp OK 15 Ex. Basc A.C. crcut h fgurs n A-group show thr typcal
More informationLie Groups HW7. Wang Shuai. November 2015
Li roups HW7 Wang Shuai Novmbr 015 1 Lt (π, V b a complx rprsntation of a compact group, show that V has an invariant non-dgnratd Hrmitian form. For any givn Hrmitian form on V, (for xampl (u, v = i u
More informationPROOF OF FIRST STANDARD FORM OF NONELEMENTARY FUNCTIONS
Intrnational Journal Of Advanc Rsarch In Scinc And Enginring http://www.ijars.com IJARSE, Vol. No., Issu No., Fbruary, 013 ISSN-319-8354(E) PROOF OF FIRST STANDARD FORM OF NONELEMENTARY FUNCTIONS 1 Dharmndra
More informationSupplementary Materials
6 Supplmntary Matrials APPENDIX A PHYSICAL INTERPRETATION OF FUEL-RATE-SPEED FUNCTION A truck running on a road with grad/slop θ positiv if moving up and ngativ if moving down facs thr rsistancs: arodynamic
More information