Hamiltonian Connectedness of Toeplitz Graphs
|
|
|
- Everett Austin
- 5 years ago
- Views:
Transcription
1 Hmiltonin Connectedne of Toeplitz Grph Muhmmd Fil Ndeem, Aeh Shir, nd Tudor Zmfirecu Introduction A imple undirected grph T with vertice ;;:::;n i clled Toeplitz grph if it djcenc mtri A.T / i Toeplitz. A Toeplitz mtri i n.n n/ mmetric mtri which h contnt vlue long ll digonl prllel to the min digonl. Therefore, Toeplitz grph T i uniquel defined the firt row of A.T /, (0 ) equence. If the in tht equence re plced t poition C t ;C t ;:::;C t k with t <t < <t k <n, we m impl write T D T n ht ;t ;:::;t k i,two vertice, eing connected n edge iff j j ft ;t ;:::;t k g. Let G e grph of order n. IticlledHmiltonin if it contin ccle of order n. It i clled trcele, if it contin pth of order n; tht pth i then clled Hmiltonin pth of G. The grph G i id to e Hmiltonin connected if for n pir of ditinct vertice u nd v of G, there eit Hmiltonin pth from u to v. The propert of eing Hmiltonin connected i tronger thn eing Hmiltonin. The third uthor work w upported grnt of the Romnin Ntionl Authorit for Scientific Reerch, CNCS UEFISCDI, project numer PN-II-ID-PCE M.F. Ndeem A. Shir Adu Slm School of Mthemticl Science, GC Univerit, 68-B, New Mulim Town, Lhore, Pkitn e-mil: mfilndeem@mil.com; hinori@hotmil.com T. Zmfirecu ( ) Fcult of Mthemtic, Univerit of Dortmund, 44 Dortmund, Germn Intitute of Mthemtic Simion Stoïlow Roumnin Acdem, Buchret, Roumni Adu Slm School of Mthemticl Science, GC Univerit, 68-B, New Mulim Town, Lhore, Pkitn e-mil: tuzmfirecu@googlemil.com. Springer Bel 05 P. Crtier et l. (ed.), Mthemtic in the t Centur, Springer Proceeding in Mthemtic & Sttitic 98, DOI 0.007/
2 36 M.F. Ndeem et l. Reference [ 4] contin reult out connectivit, iprtitene, plnrit, nd colorilit of Toeplitz grph. Some Hmiltonin propertie of undirected Toeplitz grph hve een invetigted in [] nd [5], while the directed ce w tudied in [6 8]. In [9], S. Mlik nd T. Zmfirecu trted the invetigtion of the Hmiltonin connectedne of directed Toeplitz grph. For the indirected ce, in [9] iti proven tht T n h; i i Hmiltonin connected onl for n D 3, while T n h; ; i i Hmiltonin connected for ll vlue of n nd. It will ecome cler tht, concerning k, the firt relevnt ce i k D 3. In thi pper, we re completing the picture of Hmiltonin connectedne of Toeplitz grph, more preciel of T n ht ;t i, T n h; 3; i nd T n h; 4; i. Let T e Toeplitz grph nd p; q e two vertice of T, uch tht p<q.b P p;q we men pth from p to p C contining ll vertice in fp; p C ; p C ;:::;q ; q ; qg, nd P q;p we men pth from q to q, contining the me vertice. The eitence of P p;q or P q;p i not gurnteed. We trt with few imple reult. Theorem. For n 3, T n ht ;t i i not Hmiltonin connected. Proof. Aume T D T n ht ;t i for n 3 i Hmiltonin connected. Then there eit Hmiltonin pth from t C to t C. But the pth from t C to t C contining i unique nd i of length. Thi led to contrdiction. Hence, T i not Hmiltonin connected. ut Theorem. The Toeplitz grph T n ht ;t ;t 3 ;:::; t k i i not Hmiltonin connected if t ;t ;t 3 ; :::; t k re ll odd. Proof. A iprtite grph i not Hmiltonin connected, nd if t ;t ;t 3 ; :::; t k re ll odd, then the grph T n ht ;t ;t 3 ;:::; t k i i iprtite. ut Corollr. T n h; 3; i i not Hmiltonin connected, when i odd. Theorem 3. If oth n nd t re odd, then T n h; t; n i i not Hmiltonin connected. Proof. Let, for t odd, T D T n h; t; n i, where n t C i n odd integer. Aume tht T i Hmiltonin connected, then there eit Hmiltonin pth H etween two even vertice nd of T. The pth H either contin the edge.; n/ or not. If H contin the edge.; n/, we cn contrct it to ingle verte, ecue oth vertice of the edge hve the me prit (oth re odd). After contrction, the reulting pth H 0 i of even order, nd the numer of even vertice i equl to the numer of odd vertice. But the end vertice of H 0 re even, which led to contrdiction. Net, we ume tht H doe not contin the edge.; n/. ButT without the edge.; n/ ecome T n h; ti, which i iprtite grph. Agin, H cnnot e Hmiltonin pth of T, nd thi complete the proof. ut Lemm. If n i even, then T n h; 3i dmit Hmiltonin pth from to nd, mmetr, nother one from n to n.
3 Hmiltonin Connectedne of Toeplitz Grph 37 Fig. n- n P, P,n Fig. Fig. 3 P +, P -,n Fig. 4 P -, P +,n Proof. See Fig., for Hmiltonin pth in T D T n h; 3i, from verte to verte, for even n 4. A imilr Hmiltonin pth from verte n to verte n eit in T, due to the mmetr of Toeplitz grph. Thi complete the proof. ut Lemm. Let p, q e two ditinct vertice of T n h; 3i. Ifq p i odd then pth P p;q nd P q;p eit in T n h; 3i. Proof. Appl Lemm to the ugrph of T n h; 3i pnned p; p C ;:::q. ut Theorem 4. T n h; 3; i i Hmiltonin connected for ll n C, if i n even integer. Proof. Let T D T n h; 3; i e the Toeplitz grph, where n C. Then, there eit pth P p;q nd P q;p in T, whenever q p i odd for p<q, Lemm. Now, uing uch pth of T, we prove tht for n two ditinct vertice nd of T, there eit Hmiltonin pth from to. Tke<. We plit our proof into two min ce: Ce. n i even. The following four uce rie: (i) i even, i odd. In thi ce, P ; nd P ;n eit in T, nd, with the help of thee two pth, we otin Hmiltonin pth P ; ;C ; C ;:::; ; ; P ;n in T from to ; ee Fig.. (ii) i odd, i even. If C, then poile Hmiltonin pth of T from to, PC; ;C ;:::;P ;n, i hown in Fig. 3. When D C, then for D or D n,weuethepthoflemm, nd for other vlue of, we conider the pth.;p C;n ;P ; ;/; ee Fig. 4.
4 38 M.F. Ndeem et l. P -, - P -,n Fig. 5 P +,+ + P -,n Fig. 6 Fig. 7 P,+ P +,n Fig. 8 P,+ + P,+ + P +,n Fig. 9 (iii) nd re even. If >, then Hmiltonin pth from to i.; ; P ; ; C ; :::; ; C ; C ; :::; ; P ;n / (ee Fig. 5). If, then we hve four uce to dicu:./ For > C, we conider Hmiltonin pth.; ; ;:::;;;P C;C ;C3;:::; ; P ;n / etween nd ; ee Fig. 6../ When D C n, then poile Hmiltonin pth joining nd i.; ; ;:::;;;C; P C;n ; P ;C ;CD/; ee Fig. 7..c/ If D C D n, then Hmiltonin pth from to i.; ; ;:::;3;;;P n;c /; ee Fig. 8..d/ Finll, for. A Hmiltonin pth joining nd i.; ; ;:::;;;P C;n ; ; ; 3; ; :::; C ; C ; P ;C / (ee Fig. 9). (iv) i odd, i odd. Thi ce i mmetric to ce (iii). (Denote verte i n C i.)
5 Hmiltonin Connectedne of Toeplitz Grph 39 P +,+ + P,n Fig. 0 P -,+ + P +,n Fig. P +,+ + P -,n Fig. + P +,+ P,n Fig. 3 Ce. n i odd. Agin, we conider the following uce:.i/ nd re of different prit. Firt, we ume tht <. Then Hmiltonin pth joining nd i.; ; ;:::; ; ; P ;n ; ;:::; C ; P C;C / (ee Fig. 0). If D, then Hmiltonin pth joining nd i.; ;:::;;;C; P ;C ; P C;n ;/(ee Fig. ). Net uppoe tht nd >C../ If i even, then Hmiltonin pth joining nd i.; ; ;:::;;;P C;C ;C3;:::; ; P ;n /; ee Fig.../ If i odd, then Hmiltonin pth.; ;:::;;P C;C ; C 3;:::; ; ; P ;n /, joining nd, i hown in Fig. 3. When < < nd D C, we conider Hmiltonin pth.; ;:::;4;;;3;P C;n ;P C;C / from to (ee Fig. 4).
6 40 M.F. Ndeem et l. P +, P +,n Fig. 4 P -+, -+ P -,n Fig. 5 P -+, -+ P +,n Fig. 6 P -+, P,n Fig. 7 If D nd D C, then Hmiltonin pth joining nd i. D ; ; P 3; ;P C;n ;/. Finll, here we conider the ce >../ If i even nd C, then Hmiltonin pth from to i.; ;:::; C; P C; ;C; C ;:::; ; P ;n /;ee Fig. 5. If D C, then Hmiltonin pth from to i.; P C;n ; ; ;:::; C ; P C; ;/(ee Fig. 6)../ If i odd, then Hmiltonin pth from to i.; P C; ; C ;:::; ; C ; C ;:::; ; ; P ;n /; ee Fig. 7. (ii) nd re even. The following uce rie: If D nd C,weue.;;C ;;:::;4;3;C 3; C ; C 5;:::; ; ; C ; P ;n /, the Hmiltonin pth etween nd (ee Fig. 8).
7 Hmiltonin Connectedne of Toeplitz Grph P,n Fig P,- P,n Fig. 9 P -+, +- P +-,- P,n Fig. 0 3 P -, + P +,n Fig. When 4 nd C, then Hmiltonin pth joining nd i.; C ;:::;C ; ; P ; ;C 3; C ; C 5;:::; ; ; P ;n /, hown in Fig. 9. If >,wehve.; C ; P C; ;P C; ;C 3; C ;:::; ; ; P ;n /, the Hmiltonin pth from to (ee Fig. 0). If <, the pth from to deired i.; ; 4; 5;:::;4;; C ; P C;n ;3;;5;6;:::; 3; ; C ; C ;:::; ; P ; /, when 0.mod 4/, nd.; ; 4; 5;:::;6;5;;3;P C;n ;C; ; 4; 7; 8; : : : ; 3; ; C; C;:::; ; P ; /, when.mod 4/; ee lo Fig..
8 4 M.F. Ndeem et l. Fig. P +, P -,n Fig Fig. 4 9 n n Fig n Fig. 6 (iii) nd re odd. In thi imple ce Hmiltonin pth from to i.p C; ; C ;:::; 3; ; P ;n / (ee Fig. ). Now the proof i complete. ut Lemm 3. For n D 5 nd ll n 7, T n h; 4i dmit Hmiltonin pth from to nd, mmetr, nother one from n to n. Proof. T n h; 4i i Hmiltonin for ll vlue of n ecept 6. See Fig. 3 for Hmiltonin ccle in T n h; 4i, when n f5; 7; 9g. Thee ccle re unique nd we ue them to find Hmiltonin pth from to in T n h; 4i. For n n 0.mod 3/, uitle pth i otined uing the Hmiltonin ccle in T 9 h; 4i; ee Fig. 4. To otin uch pth when n.mod 3/, we ue the Hmiltonin ccle found in T 7 h; 4i; ee Fig. 5. For n.mod 3/, the ccle T 5 h; 4i i emploed; ee Fig. 6. Now, ecue of the mmetr of the Toeplitz grph, we lo hve Hmiltonin pth from n to n. ut
9 Hmiltonin Connectedne of Toeplitz Grph 43 P -, - P,n P, + P +,n Fig P 4,n P 4,n c P 6,n Fig. 8 Lemm 4. Let p, q e two ditinct vertice of T n h; 4i. Ifq p ; 3; 5, then there eit pth P p;q nd P q;p in T n h; 4i. Proof. See Lemm 3. ut Theorem 5. T n h; 4; i i Hmiltonin connected for ll nd n 5. Proof. For n 5; let nd e ditinct vertice of the Toeplitz grph T D T n h; 4; i. Aume tht <. To prove the reult we how tht there eit Hmiltonin pth etween nd. Ce. D C. If D or n, we hve deired pth due to Lemm 3. When 5 n 5, then Hmiltonin pth etween nd i either.; P ; ;P ;n / or.p ; ;P C;n ;/(ee Fig. 7). When 4, ee Fig. 8 for Hmiltonin pth etween nd. For n 4 n, the deired Hmiltonin pth re mmetric to the pth for f; 3; 4g. Ce. If C, the following three uce rie: (i) n 5 nd f3;4;:::;n 5; n 3g. (ii) n 5 nd fn 4; n ; n ; ng. (iii) n 4. Suce (i). Let f3;4;:::;n 5; n 3g../ Firt, we ume the ce when f4;6;7;:::;n 5g.Now.P C; ;C ; C 3; :::; ; P ;n / i required Hmiltonin pth etween nd (ee Fig. 9)../ If D, then deired pth etween nd i hown in Fig. 30.
10 44 M.F. Ndeem et l. P +, + - P -,n Fig. 9 Fig. 30 P -, P 6,n 7 P 6,n c d P 9,n P 9,n e P -, Fig. 3 () A Hmiltonin pth etween nd 4. () A Hmiltonin pth etween nd 6. (c) A Hmiltonin pth etween nd 7. (d) A Hmiltonin pth etween nd 8. (e) A Hmiltonin pth etween nd, where P,n Fig. 3.c/ If D nd 5, then Hmiltonin pth etween nd different vlue of re hown in Fig. 3. When D nd D 5, to get deired pth, we ue the difference long with difference nd 4. See Fig. 3, for uch pth when f8; 9; 0; : : : ; n 6; n 4g. When D 5; 6; 7, ee Fig. 33.
11 Hmiltonin Connectedne of Toeplitz Grph P 7,n P 9,n c P 9,n Fig. 33 () D 5. () D 6. (c) D n-4 n n- n c d n- n n Fig. 34 () D n 5. () D n 3. (c) D n. (d) D n P 7,n P 7,n c P -,n Fig. 35 () A Hmiltonin pth etween 3 nd 5. () A Hmiltonin pth etween 3 nd 6. (c) A Hmiltonin pth etween 3 nd 7 And, for D n 5; n 3; n ; n, ee Fig. 34.d/ If D 3, then for Hmiltonin pth etween 3 nd, ee Fig. 35..e/ If D 5 nd 8, deired Hmiltonin pth i hown in Fig. 36. When D 8 nd n 5; 7, we ue the pth hown in Fig. 37. For n D 5 nd n D 7, ee Fig. 38 nd 39, repectivel.
12 46 M.F. Ndeem et l P 9,n P -,n Fig. 36 () A Hmiltonin pth etween 5 nd 7.() A Hmiltonin pth etween 5 nd,where P,n Fig c d e f g h i j Fig. 38 Hmiltonin pth etween 5 nd 8 for different vlue of,whenn D 5.() D 5.() D 6. (c) D 7. (d) D 8.(e) D 9. (f) D 0. (g) D. (h) D. (i) D 3. (j) D 4
13 Hmiltonin Connectedne of Toeplitz Grph c d e f g h i j k l Fig. 39 Hmiltonin pth etween 5 nd 8 for different vlue of,whenn D 7.() D 5.() D 6.(c) D 7.(d) D 8.(e) D 9.(f) D 0.(g) D.(h) D.(i) D 3.(j) D 4. (k) D 5. (l) D 6 Suce (ii). Thi uce i mmetricl to f; ; 3; 5g nd 6.Itw treted inide of.i/ ecept for the ce D n 4; n ; n ; n. To otin Hmiltonin pth from f; ; 3; 5g to fn 4; n ; n ; ng, we firt collect the four Hmiltonin pth in T 8 h; 4i from f; ; 3; 5g to 8; ee Fig. 40. Smmetricll, we hve pth in T n h; 4i from fn 4; n ; n ; ng to n 7, of verte et fn 7; n 6;:::;ng. Joining 8ton 7thedirectpth(8,9,...,n 7) give the deired Hmiltonin pth in T n h; 4i from to. Suce (iii). Thi uce i mmetricl with 5, treted inide of.i/. ut
14 48 M.F. Ndeem et l Fig. 40 To ee whether T n h; 4; i i Hmiltonin connected or not, for 6 n 4, ee the following tle: Hmiltonin connected when i T 6 h; 4; i T 7 h; 4; i T 8 h; 4; i 5, 7 T 9 h; 4; i 5, 8 T 0 h; 4; i 5, 6, 7, 9 T h; 4; i 5, 7, 8, 0 T h; 4; i 5, 6, 7, 8, 9, T 3 h; 4; i for ll T 4 h; 4; i 5, 6, 7, 8, 9, 0,, 3 Miing vlue for men tht the correponding Toeplitz grph i not Hmiltonin connected. Thi w verified uing computer. Reference. vn Dl, R., Tijen, G., Tuz, Z., vn der Veen, J.A.A., Zmfirecu, Ch., Zmfirecu, T.: Hmiltonin propertie of Toeplitz grph. Dicret. Mth. 59, 69 8 (996). Euler, R.: Chrcterizing iprtite Toeplitz grph. Theor. Comput. Sci. 63, (00) 3. Euler, R., LeVerge, H., Zmfirecu, T.: A chrcteriztion of infinite, iprtite Toeplitz grph. In: Tung-Hin, K. (ed.) Comintoric nd Grph Theor 95, Vol.. Acdemi Sinic, pp World Scientific, Singpore (995) 4. Euler, R., Zmfirecu, T.: On plnr Toeplitz grph. Grph Com. 9, 3 37 (03) 5. Heuerger, C.: On Hmiltonin Toeplitz grph. Dicret. Mth. 45, 07 5 (00) 6. Mlik, S.: Hmiltonin ccle in directed Toeplitz grph II. Ar Com. (to pper)
15 Hmiltonin Connectedne of Toeplitz Grph Mlik, S.: Hmiltonicit in directed Toeplitz grph of mimum (out or in) degree 4. Util. Mth. 89, (0) 8. Mlik, S., Qurehi, A.M.: Hmiltonin ccle in directed Toeplitz grph. Ar Com. 09, 5 56 (03) 9. Mlik, S., Zmfirecu, T.: Hmiltonin connectedne in directed Toeplitz grph. Bull. Mth. Soc. Sci. Mth. Roum. 53(0) No., (00)
EE Control Systems LECTURE 8
Coyright F.L. Lewi 999 All right reerved Udted: Sundy, Ferury, 999 EE 44 - Control Sytem LECTURE 8 REALIZATION AND CANONICAL FORMS A liner time-invrint (LTI) ytem cn e rereented in mny wy, including: differentil
20.2. The Transform and its Inverse. Introduction. Prerequisites. Learning Outcomes
The Trnform nd it Invere 2.2 Introduction In thi Section we formlly introduce the Lplce trnform. The trnform i only pplied to cul function which were introduced in Section 2.1. We find the Lplce trnform
Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
Math 2142 Homework 2 Solutions. Problem 1. Prove the following formulas for Laplace transforms for s > 0. a s 2 + a 2 L{cos at} = e st.
Mth 2142 Homework 2 Solution Problem 1. Prove the following formul for Lplce trnform for >. L{1} = 1 L{t} = 1 2 L{in t} = 2 + 2 L{co t} = 2 + 2 Solution. For the firt Lplce trnform, we need to clculte:
Answers and Solutions to (Some Even Numbered) Suggested Exercises in Chapter 11 of Grimaldi s Discrete and Combinatorial Mathematics
Answers n Solutions to (Some Even Numere) Suggeste Exercises in Chpter 11 o Grimli s Discrete n Comintoril Mthemtics Section 11.1 11.1.4. κ(g) = 2. Let V e = {v : v hs even numer o 1 s} n V o = {v : v
Proceedings of the International Conference on Theory and Applications of Mathematics and Informatics ICTAMI 2003, Alba Iulia
Proceedings o the Interntionl Conerence on Theor nd Applictions o Mthemtics nd Inormtics ICTAMI 2003, Al Iuli CARACTERIZATIONS OF TE FUNCTIONS WIT BOUNDED VARIATION Dniel Lesnic Astrct. The present stud
P 1 (x 1, y 1 ) is given by,.
MA00 Clculus nd Bsic Liner Alger I Chpter Coordinte Geometr nd Conic Sections Review In the rectngulr/crtesin coordintes sstem, we descrie the loction of points using coordintes. P (, ) P(, ) O The distnce
Solutions Problem Set 2. Problem (a) Let M denote the DFA constructed by swapping the accept and non-accepting state in M.
Solution Prolem Set 2 Prolem.4 () Let M denote the DFA contructed y wpping the ccept nd non-ccepting tte in M. For ny tring w B, w will e ccepted y M, tht i, fter conuming the tring w, M will e in n ccepting
Continuity. Recall the following properties of limits. Theorem. Suppose that lim. f(x) =L and lim. lim. [f(x)g(x)] = LM, lim
![Continuity. Recall the following properties of limits. Theorem. Suppose that lim. f(x) =L and lim. lim. [f(x)g(x)] = LM, lim](/thumbs/72/66328186.jpg "Continuity. Recall the following properties of limits. Theorem. Suppose that lim. f(x) =L and lim. lim. [f(x)g(x)] = LM, lim")
Recll the following properties of limits. Theorem. Suppose tht lim f() =L nd lim g() =M. Then lim [f() ± g()] = L + M, lim [f()g()] = LM, if M = 0, lim f() g() = L M. Furthermore, if f() g() for ll, then
On the Adders with Minimum Tests
Proceeding of the 5th Ain Tet Sympoium (ATS '97) On the Adder with Minimum Tet Seiji Kjihr nd Tutomu So Dept. of Computer Science nd Electronic, Kyuhu Intitute of Technology Atrct Thi pper conider two
Hamiltonian Cycle in Complete Multipartite Graphs
Annls of Pure nd Applied Mthemtics Vol 13, No 2, 2017, 223-228 ISSN: 2279-087X (P), 2279-0888(online) Pulished on 18 April 2017 wwwreserchmthsciorg DOI: http://dxdoiorg/1022457/pmv13n28 Annls of Hmiltonin
2. The Laplace Transform
. The Lplce Trnform. Review of Lplce Trnform Theory Pierre Simon Mrqui de Lplce (749-87 French tronomer, mthemticin nd politicin, Miniter of Interior for 6 wee under Npoleon, Preident of Acdemie Frncie
PHYSICS 211 MIDTERM I 22 October 2003
PHYSICS MIDTERM I October 3 Exm i cloed book, cloed note. Ue onl our formul heet. Write ll work nd nwer in exm booklet. The bck of pge will not be grded unle ou o requet on the front of the pge. Show ll
ARCHIVUM MATHEMATICUM (BRNO) Tomus 47 (2011), Kristína Rostás
ARCHIVUM MAHEMAICUM (BRNO) omu 47 (20), 23 33 MINIMAL AND MAXIMAL SOLUIONS OF FOURH ORDER IERAED DIFFERENIAL EQUAIONS WIH SINGULAR NONLINEARIY Kritín Rotá Abtrct. In thi pper we re concerned with ufficient
Primitive Digraphs with the Largest Scrambling Index
Primitive Digraph with the Larget Scrambling Index Mahmud Akelbek, Steve Kirkl 1 Department of Mathematic Statitic, Univerity of Regina, Regina, Sakatchewan, Canada S4S 0A Abtract The crambling index of
ON ALTERNATING POWER SUMS OF ARITHMETIC PROGRESSIONS
ON ALTERNATING POWER SUMS OF ARITHMETIC PROGRESSIONS A. BAZSÓ Astrct. Depending on the prity of the positive integer n the lternting power sum T k n = k + k + + k...+ 1 n 1 n 1 + k. cn e extended to polynomil
Binding 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.
Kirchhoff index of graphs and some graph operations
Proc Indin Acd Sci (Mth Sci) Vol 124, No 3, August 2014, pp 281 289 c Indin Acdem of Sciences Kirchhoff inde of grphs nd some grph opertions A NIKSERESHT, Z SEPASDAR nd M H SHIRDAREH-HAGHIGHI Deprtment
Zero-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
p-adic Egyptian Fractions
p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction
Convex Hulls of Curves Sam Burton
Convex Hull of Curve Sam Burton 1 Introduction Thi paper will primarily be concerned with determining the face of convex hull of curve of the form C = {(t, t a, t b ) t [ 1, 1]}, a < b N in R 3. We hall
CHAPTER 8 OBSERVER BASED REDUCED ORDER CONTROLLER DESIGN FOR LARGE SCALE LINEAR DISCRETE-TIME CONTROL SYSTEMS
CHAPTER 8 OBSERVER BASED REDUCED ORDER CONTROLLER DESIGN FOR LARGE SCALE LINEAR DISCRETE-TIME CONTROL SYSTEMS 8.1 INTRODUCTION 8.2 REDUCED ORDER MODEL DESIGN FOR LINEAR DISCRETE-TIME CONTROL SYSTEMS 8.3
Bases 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
On 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
2.4 Linear Inequalities and Interval Notation
.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or
Tutorial 2 Euler Lagrange ( ) ( ) In one sentence: d dx
Tutoril 2 Euler Lgrnge In one entene: d Fy = F d Importnt ft: ) The olution of EL eqution i lled eterml. 2) Minmum / Mimum of the "Mot Simple prolem" i lo n eterml. 3) It i eier to olve EL nd hek if we
Analysis of Variance and Design of Experiments-II
Anlyi of Vrince nd Deign of Experiment-II MODULE VI LECTURE - 7 SPLIT-PLOT AND STRIP-PLOT DESIGNS Dr. Shlbh Deprtment of Mthemtic & Sttitic Indin Intitute of Technology Knpur Anlyi of covrince ith one
Graph Theory. Simple Graph G = (V, E). V={a,b,c,d,e,f,g,h,k} E={(a,b),(a,g),( a,h),(a,k),(b,c),(b,k),...,(h,k)}
Grph Theory Simple Grph G = (V, E). V ={verties}, E={eges}. h k g f e V={,,,,e,f,g,h,k} E={(,),(,g),(,h),(,k),(,),(,k),...,(h,k)} E =16. 1 Grph or Multi-Grph We llow loops n multiple eges. G = (V, E.ψ)
Section The Precise Definition Of A Limit
Section 2.4 - The Precise Definition Of A imit Introduction So fr we hve tken n intuitive pproch to the concept of limit. In this section we will stte the forml definition nd use this definition to prove
COUNTING DESCENTS, RISES, AND LEVELS, WITH PRESCRIBED FIRST ELEMENT, IN WORDS
COUNTING DESCENTS, RISES, AND LEVELS, WITH PRESCRIBED FIRST ELEMENT, IN WORDS Sergey Kitev The Mthemtic Intitute, Reykvik Univerity, IS-03 Reykvik, Icelnd ergey@rui Toufik Mnour Deprtment of Mthemtic,
Coalgebra, Lecture 15: Equations for Deterministic Automata
Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined
4-4 E-field Calculations using Coulomb s Law
1/11/5 ection_4_4_e-field_clcultion_uing_coulomb_lw_empty.doc 1/1 4-4 E-field Clcultion uing Coulomb Lw Reding Aignment: pp. 9-98 Specificlly: 1. HO: The Uniform, Infinite Line Chrge. HO: The Uniform Dik
Mathematics. Area under Curve.
Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding
, MATHS H.O.D.: SUHAG R.KARIYA, BHOPAL, CONIC SECTION PART 8 OF
DOWNLOAD FREE FROM www.tekoclsses.com, PH.: 0 903 903 7779, 98930 5888 Some questions (Assertion Reson tpe) re given elow. Ech question contins Sttement (Assertion) nd Sttement (Reson). Ech question hs
MATH4455 Module 10 Evens, Odds and Ends
MATH4455 Module 10 Even, Odd nd End Min Mth Conept: Prity, Ple Vlue Nottion, Enumertion, Story Prolem Auxiliry Ide: Tournment, Undireted grph I. The Mind-Reding Clultor Prolem. How doe the mind-reding
Homework Solution - Set 5 Due: Friday 10/03/08
CE 96 Introduction to the Theory of Computtion ll 2008 Homework olution - et 5 Due: ridy 10/0/08 1. Textook, Pge 86, Exercise 1.21. () 1 2 Add new strt stte nd finl stte. Mke originl finl stte non-finl.
Chapter 1: Logarithmic functions and indices
Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4
Transfer Functions. Chapter 5. Transfer Functions. Derivation of a Transfer Function. Transfer Functions
5/4/6 PM : Trnfer Function Chpter 5 Trnfer Function Defined G() = Y()/U() preent normlized model of proce, i.e., cn be ued with n input. Y() nd U() re both written in devition vrible form. The form of
First Midterm Examination
Çnky University Deprtment of Computer Engineering 203-204 Fll Semester First Midterm Exmintion ) Design DFA for ll strings over the lphet Σ = {,, c} in which there is no, no nd no cc. 2) Wht lnguge does
If C = 60 and = P, find the value of P. c 2 = a 2 + b 2 2abcos 60 = a 2 + b 2 ab a 2 + b 2 = c 2 + ab. c a
Answers: (000-0 HKMO Finl Events) Creted : Mr. Frncis Hung Lst updted: 0 June 08 Individul Events I P I P I P I P 5 7 0 0 S S S S Group Events G G G G 80 00 0 c 8 c c c d d 6 d 5 d 85 Individul Event I.,
Lyapunov-type inequality for the Hadamard fractional boundary value problem on a general interval [a; b]; (1 6 a < b)
![Lyapunov-type inequality for the Hadamard fractional boundary value problem on a general interval [a; b]; (1 6 a < b)](/thumbs/91/105870616.jpg "Lyapunov-type inequality for the Hadamard fractional boundary value problem on a general interval [a; b]; (1 6 a < b)")
Lypunov-type inequlity for the Hdmrd frctionl boundry vlue problem on generl intervl [; b]; ( 6 < b) Zid Ldjl Deprtement of Mthemtic nd Computer Science, ICOSI Lbortory, Univerity of Khenchel, 40000, Algeri.
Topics Covered AP Calculus AB
Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.
Research Article Generalized Hyers-Ulam Stability of the Second-Order Linear Differential Equations
Hindwi Publihing Corportion Journl of Applied Mthemtic Volume 011, Article ID 813137, 10 pge doi:10.1155/011/813137 Reerch Article Generlized Hyer-Ulm Stbility of the Second-Order Liner Differentil Eqution
378 Relations Solutions for Chapter 16. Section 16.1 Exercises. 3. Let A = {0,1,2,3,4,5}. Write out the relation R that expresses on A.
378 Reltions 16.7 Solutions for Chpter 16 Section 16.1 Exercises 1. Let A = {0,1,2,3,4,5}. Write out the reltion R tht expresses > on A. Then illustrte it with digrm. 2 1 R = { (5,4),(5,3),(5,2),(5,1),(5,0),(4,3),(4,2),(4,1),
First Midterm Examination
24-25 Fll Semester First Midterm Exmintion ) Give the stte digrm of DFA tht recognizes the lnguge A over lphet Σ = {, } where A = {w w contins or } 2) The following DFA recognizes the lnguge B over lphet
Some circular summation formulas for theta functions
Ci et l. Boundr Vlue Prolems 013, 013:59 R E S E A R C H Open Access Some circulr summtion formuls for thet functions Yi Ci, Si Chen nd Qiu-Ming Luo * * Correspondence: luomth007@163.com Deprtment of Mthemtics,
Thomas Whitham Sixth Form
Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos
ON THE NILPOTENCY INDEX OF THE RADICAL OF A GROUP ALGEBRA. XI
Mth. J. Okym Univ. 44(2002), 51 56 ON THE NILPOTENCY INDEX OF THE RADICAL OF A GROUP ALGEBRA. XI Koru MOTOSE Let t(g) be the nilpotency index of the rdicl J(KG) of group lgebr KG of finite p-solvble group
CIRCULAR COLOURING THE PLANE
CIRCULAR COLOURING THE PLANE MATT DEVOS, JAVAD EBRAHIMI, MOHAMMAD GHEBLEH, LUIS GODDYN, BOJAN MOHAR, AND REZA NASERASR Astrct. The unit distnce grph R is the grph with vertex set R 2 in which two vertices
Binding Numbers for all Fractional (a, b, k)-critical Graphs
Filomt 28:4 (2014), 709 713 DOI 10.2298/FIL1404709Z Published by Fculty of Sciences nd Mthemtics, University of Niš, Serbi Avilble t: http://www.pmf.ni.c.rs/filomt Binding Numbers for ll Frctionl (, b,
8.6 The Hyperbola. and F 2. is a constant. P F 2. P =k The two fixed points, F 1. , are called the foci of the hyperbola. The line segments F 1
8. The Hperol Some ships nvigte using rdio nvigtion sstem clled LORAN, which is n cronm for LOng RAnge Nvigtion. A ship receives rdio signls from pirs of trnsmitting sttions tht send signls t the sme time.
A New Solution to the Intersection Problem of Mendelsohn Triple Systems. Rachel Watson
A New Solution to the Intersection Prolem of Mendelsohn Triple Systems y Rchel Wtson A thesis sumitted to the Grdute Fculty of Auurn University in prtil fulfillment of the requirements for the Degree of
QUADRATIC EQUATIONS OBJECTIVE PROBLEMS
QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.
Torsion in Groups of Integral Triangles
Advnces in Pure Mthemtics, 01,, 116-10 http://dxdoiorg/1046/pm011015 Pulished Online Jnury 01 (http://wwwscirporg/journl/pm) Torsion in Groups of Integrl Tringles Will Murry Deprtment of Mthemtics nd Sttistics,
Duke Math Meet
Duke Mth Meet 01-14 Power Round Qudrtic Residues nd Prime Numers For integers nd, we write to indicte tht evenly divides, nd to indicte tht does not divide For exmle, 4 nd 4 Let e rime numer An integer
M344 - ADVANCED ENGINEERING MATHEMATICS
M3 - ADVANCED ENGINEERING MATHEMATICS Lecture 18: Lplce s Eqution, Anltic nd Numericl Solution Our emple of n elliptic prtil differentil eqution is Lplce s eqution, lso clled the Diffusion Eqution. If
Math 4310 Solutions to homework 1 Due 9/1/16
Mth 4310 Solutions to homework 1 Due 9/1/16 1. Use the Eucliden lgorithm to find the following gretest common divisors. () gcd(252, 180) = 36 (b) gcd(513, 187) = 1 (c) gcd(7684, 4148) = 68 252 = 180 1
Decomposition of terms in Lucas sequences
Journl of Logic & Anlysis 1:4 009 1 3 ISSN 1759-9008 1 Decomposition of terms in Lucs sequences ABDELMADJID BOUDAOUD Let P, Q be non-zero integers such tht D = P 4Q is different from zero. The sequences
SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics
SCHOOL OF ENGINEERING & BUIL ENVIRONMEN Mthemtics An Introduction to Mtrices Definition of Mtri Size of Mtri Rows nd Columns of Mtri Mtri Addition Sclr Multipliction of Mtri Mtri Multipliction 7 rnspose
CHOOSING THE NUMBER OF MODELS OF THE REFERENCE MODEL USING MULTIPLE MODELS ADAPTIVE CONTROL SYSTEM
Interntionl Crpthin Control Conference ICCC 00 ALENOVICE, CZEC REPUBLIC y 7-30, 00 COOSING TE NUBER OF ODELS OF TE REFERENCE ODEL USING ULTIPLE ODELS ADAPTIVE CONTROL SYSTE rin BICĂ, Victor-Vleriu PATRICIU
Fundamental Theorem of Calculus
Fundmentl Theorem of Clculus Recll tht if f is nonnegtive nd continuous on [, ], then the re under its grph etween nd is the definite integrl A= f() d Now, for in the intervl [, ], let A() e the re under
( β ) touches the x-axis if = 1
Generl Certificte of Eduction (dv. Level) Emintion, ugust Comined Mthemtics I - Prt B Model nswers. () Let f k k, where k is rel constnt. i. Epress f in the form( ) Find the turning point of f without
Existence and Uniqueness of Solution for a Fractional Order Integro-Differential Equation with Non-Local and Global Boundary Conditions
Applied Mthetic 0 9-96 doi:0.436/.0.079 Pulihed Online Octoer 0 (http://www.scirp.org/journl/) Eitence nd Uniquene of Solution for Frctionl Order Integro-Differentil Eqution with Non-Locl nd Glol Boundry
Parallel Projection Theorem (Midpoint Connector Theorem):
rllel rojection Theorem (Midpoint onnector Theorem): The segment joining the midpoints of two sides of tringle is prllel to the third side nd hs length one-hlf the third side. onversely, If line isects
IEOR 3106: Fall 2013, Professor Whitt Topics for Discussion: Tuesday, November 19 Alternating Renewal Processes and The Renewal Equation
IEOR 316: Fall 213, Profeor Whitt Topic for Dicuion: Tueday, November 19 Alternating Renewal Procee and The Renewal Equation 1 Alternating Renewal Procee An alternating renewal proce alternate between
arxiv: v1 [math.co] 8 Sep 2015
![arxiv: v1 [math.co] 8 Sep 2015](/thumbs/73/69196870.jpg "arxiv: v1 [math.co] 8 Sep 2015")
Accepted for puliction in Mthemtic Slovc ZERO-DIVISOR GRAPHS OF LOWER DISMANTLABLE LATTICES-I AVINASH PATIL*, B. N. WAPHARE**, V. V. JOSHI** AND H. Y. POURALI** Dedicted to Professor N. K. Thkre on his
8 factors of x. For our second example, let s raise a power to a power:
CH 5 THE FIVE LAWS OF EXPONENTS EXPONENTS WITH VARIABLES It s no time for chnge in tctics, in order to give us deeper understnding of eponents. For ech of the folloing five emples, e ill stretch nd squish,
Problem Set 8 Solutions
Deign and Analyi of Algorithm April 29, 2015 Maachuett Intitute of Technology 6.046J/18.410J Prof. Erik Demaine, Srini Devada, and Nancy Lynch Problem Set 8 Solution Problem Set 8 Solution Thi problem
Signal Flow Graphs. Consider a complex 3-port microwave network, constructed of 5 simpler microwave devices:
3/3/009 ignl Flow Grphs / ignl Flow Grphs Consider comple 3-port microwve network, constructed of 5 simpler microwve devices: 3 4 5 where n is the scttering mtri of ech device, nd is the overll scttering
The Shortest Path Problem Graph Algorithms - 3
Algorithm Deign nd Anli Vitor Admhik C - pring Leture Feb, Crnegie Mellon Univerit The hortet Pth Problem Grph Algorithm - The hortet Pth Problem Given poitivel weighted grph G with oure verte, find the
PART - III : MATHEMATICS
JEE(Advnced) 4 Finl Em/Pper-/Code-8 PART - III : SECTION : (One or More Thn One Options Correct Type) This section contins multiple choice questions. Ech question hs four choices (A), (B), (C) nd (D) out
Suppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = -2.
Mth 43 Section 6. Section 6.: Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot
arxiv: v1 [math.mg] 25 Aug 2011
![arxiv: v1 [math.mg] 25 Aug 2011](/thumbs/72/67862976.jpg "arxiv: v1 [math.mg] 25 Aug 2011")
ABSORBING ANGLES, STEINER MINIMAL TREES, AND ANTIPODALITY HORST MARTINI, KONRAD J. SWANEPOEL, AND P. OLOFF DE WET arxiv:08.5046v [math.mg] 25 Aug 20 Abtract. We give a new proof that a tar {op i : i =,...,
Graph 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
GUC (Dr. Hany Hammad) 9/19/2016
UC (Dr. Hny Hmmd) 9/9/6 ecture # ignl flw grph: Defitin. Rule f Reductin. Mn Rule. ignl-flw grph repreenttin f : ltge urce. ive gle-prt device. ignl Flw rph A ignl-flw grph i grphicl men f prtryg the reltinhip
π = tanc 1 + tan x ...(i)
Solutions to RSPL/ π. Let, I log ( tn ) d Using f () d f ( ) d π π I log( tnc d m log( cot ) d...(ii) On dding (i) nd (ii), we get +,. Given f() + ), For continuit t lim " lim f () " ( ) \ Continuous t.
N 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
KRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION
Fixed Point Theory, 13(2012), No. 1, 285-291 http://www.mth.ubbcluj.ro/ nodecj/sfptcj.html KRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION FULI WANG AND FENG WANG School of Mthemtics nd
Lecture 8: Period Finding: Simon s Problem over Z N
Quantum Computation (CMU 8-859BB, Fall 205) Lecture 8: Period Finding: Simon Problem over Z October 5, 205 Lecturer: John Wright Scribe: icola Rech Problem A mentioned previouly, period finding i a rephraing
Rectangular group congruences on an epigroup
cholrs Journl of Engineering nd Technology (JET) ch J Eng Tech, 015; 3(9):73-736 cholrs Acdemic nd cientific Pulisher (An Interntionl Pulisher for Acdemic nd cientific Resources) wwwsspulishercom IN 31-435X
Counting intersections of spirals on a torus
Counting intersections of spirls on torus 1 The problem Consider unit squre with opposite sides identified. For emple, if we leve the centre of the squre trveling long line of slope 2 (s shown in the first
Mrgolius 2 In the rticulr cse of Ploue's constnt, we tke = 2= 5+i= 5, nd = ;1, then ; C = tn;1 1 2 = ln(2= 5+i= 5) ln(;1) More generlly, we would hve
Ploue's Constnt is Trnscendentl Brr H. Mrgolius Clevelnd Stte University Clevelnd, Ohio 44115.mrgolius@csuohio.edu Astrct Ploue's constnt tn;1 ( 1 2) is trnscendentl. We rove this nd more generl result
Acta Universitatis Carolinae. Mathematica et Physica
Act Universittis Croline. Mthemtic et Physic Thoms N. Vougiouklis Cyclicity in specil clss of hypergroups Act Universittis Croline. Mthemtic et Physic, Vol. 22 (1981), No. 1, 3--6 Persistent URL: http://dml.cz/dmlcz/142458
10.2 The Ellipse and the Hyperbola
CHAPTER 0 Conic Sections Solve. 97. Two surveors need to find the distnce cross lke. The plce reference pole t point A in the digrm. Point B is meters est nd meter north of the reference point A. Point
Here we study square linear systems and properties of their coefficient matrices as they relate to the solution set of the linear system.
Section 24 Nonsingulr Liner Systems Here we study squre liner systems nd properties of their coefficient mtrices s they relte to the solution set of the liner system Let A be n n Then we know from previous
1. A refinement of a classical class number relation We give a refinement, and a new proof, of the following classical result [1, 2, 3].
![1. A refinement of a classical class number relation We give a refinement, and a new proof, of the following classical result [1, 2, 3].](/thumbs/78/78385403.jpg "1. A refinement of a classical class number relation We give a refinement, and a new proof, of the following classical result [1, 2, 3].")
A COMBINATORIAL REFINEMENT OF THE KRONECKER-HURWITZ CLASS NUMBER RELATION rxiv:604.08v [mth.nt] Apr 06 ALEXANDRU A. POPA AND DON ZAGIER Astrct. We give refinement of the Kronecker-Hurwitz clss numer reltion,
Position. If the particle is at point (x, y, z) on the curved path s shown in Fig a,then its location is defined by the position vector
34 C HAPTER 1 KINEMATICS OF A PARTICLE 1 1.5 Curvilinear Motion: Rectangular Component Occaionall the motion of a particle can bet be decribed along a path that can be epreed in term of it,, coordinate.
Quadratic Residues. Chapter Quadratic residues
Chter 8 Qudrtic Residues 8. Qudrtic residues Let n>be given ositive integer, nd gcd, n. We sy tht Z n is qudrtic residue mod n if the congruence x mod n is solvble. Otherwise, is clled qudrtic nonresidue
WENJUN LIU AND QUÔ C ANH NGÔ
AN OSTROWSKI-GRÜSS TYPE INEQUALITY ON TIME SCALES WENJUN LIU AND QUÔ C ANH NGÔ Astrct. In this pper we derive new inequlity of Ostrowski-Grüss type on time scles nd thus unify corresponding continuous
Exercise sheet 6: Solutions
Eerise sheet 6: Solutions Cvet emptor: These re merel etended hints, rther thn omplete solutions. 1. If grph G hs hromti numer k > 1, prove tht its verte set n e prtitioned into two nonempt sets V 1 nd
(e) if x = y + z and a divides any two of the integers x, y, or z, then a divides the remaining integer
Divisibility In this note we introduce the notion of divisibility for two integers nd b then we discuss the division lgorithm. First we give forml definition nd note some properties of the division opertion.
1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the
LINEAR ALGEBRA APPLIED
5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nth-order
List all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1.
Mth Anlysis CP WS 4.X- Section 4.-4.4 Review Complete ech question without the use of grphing clcultor.. Compre the mening of the words: roots, zeros nd fctors.. Determine whether - is root of 0. Show
Mathematical Sciences Technical Reports (MSTR)
Roe-Hulmn Intitute of Technology Roe-Hulmn Scholr Mthemticl Science Technicl Report (MSTR) Mthemtic 8-15-9 Flttening Cone Sen A. Broughton Roe-Hulmn Intitute of Technology, brought@roe-hulmn.edu Follow
ISEM Team-Lecce. holds for some c > 0 and all y Y. Assume T (t)y Y for all t 0 and T ( )y C(R +, Y ) for all y Y. For t 0 we define the operators
ISEM Tem-Lecce EXERCISE 3.. Let A generte the C -semigroup T ( ) on Bnch spce X. Let J : X E be n isomorphism to nother Bnch spce E, Y X be Bnch subspce which is equipped with norm Y such tht X c Y holds
On rational Diophantine Triples and Quadruples
Interntionl Journl of Scientific nd Reserch Pulictions, Volume 4, Issue 9, Septemer 014 1 ISSN 50-3153 On rtionl Diophntine riples nd Qudruples M.A.Gopln *, K.Geeth **, Mnju Somnth *** * Professor,Dept.of
1 2 : 4 5. Why Digital Systems? Lesson 1: Introduction to Digital Logic Design. Numbering systems. Sample Problems 1 5 min. Lesson 1-b: Logic Gates
Leon : Introduction to Digitl Logic Deign Computer ided Digitl Deign EE 39 meet Chvn Fll 29 Why Digitl Sytem? ccurte depending on numer of digit ued CD Muic i digitl Vinyl Record were nlog DVD Video nd
THE DIVERGENCE-FREE JACOBIAN CONJECTURE IN DIMENSION TWO
THE DIVERGENCE-FREE JACOBIAN CONJECTURE IN DIMENSION TWO J. W. NEUBERGER Abtract. A pecial cae, called the divergence-free cae, of the Jacobian Conjecture in dimenion two i proved. Thi note outline an
Math 131. Numerical Integration Larson Section 4.6
Mth. Numericl Integrtion Lrson Section. This section looks t couple of methods for pproimting definite integrls numericlly. The gol is to get good pproimtion of the definite integrl in problems where n