SOME NEW SELF-DUAL [96, 48, 16] CODES WITH AN AUTOMORPHISM OF ORDER 15. KEYWORDS: automorphisms, construction, self-dual codes
|
|
- Silas Andrews
- 6 years ago
- Views:
Transcription
1 Факултет по математика и информатика, том ХVІ С, 014 SOME NEW SELF-DUAL [96, 48, 16] CODES WITH AN AUTOMORPHISM OF ORDER 15 NIKOLAY I. YANKOV ABSTRACT: A new method fo constuctng bnay self-dual codes wth an automophsm of ode pq fo p q was developed n [1]. We use ths method to constuct new doubly-even self-dual [96, 48, 16] codes havng an automophsm of ode 15 wth 6 cycles of length 15 and two cycles of length 3. Moe than new such codes ae obtan. We found exactly 19 dffeent values of the paamete n the weght dstbuton of these codes of whch 11 ae new. KEYWORDS: automophsms, constucton, self-dual codes 1. Intoducton A lnea [ n, k] code C s a k-dmensonal subspace of the vecto space GF( q ) n, whee GF( q ) s the fnte feld of q elements. The elements of C ae called codewods and the (Hammng) weght of a codewod s the numbe of ts nonzeo coodnate postons. The mnmum weght d of C s the smallest weght among all nonzeo codewods of C, and C s called a [ n, k, d] code. A matx whch ows fom a bass of C s called the geneato matx of ths code. The weght enumeato W (y) of a code C s gven by W n ( y ) = A y = 0 whee A, s the numbe of codewods of n n weght n C. Let ( u, v) : Fq Fq Fq be an nne poduct n the lnea space F. The dual code of C s C n q n = { u F : ( u, v) = 0 fo all v C}. The dual code q C s a Ths wok s suppoted by Shumen Unvesty unde Poect RD-08-34/
2 Факултет по математика и информатика, том ХVІ С, 014 lnea [ n, n k] code. We call the code C self-othogonal f C C. If C = C then the code C s temed self-dual. Two bnay codes ae equvalent f one can be obtaned fom the othe by a pemutaton of coodnates. The pemutaton σ S n s an automophsm of C, f C = σ (C). The set of all automophsms of C foms a goup, called the automophsm goup Aut (C) of C. Fo two dffeent pmes p < q we say that an automophsm σ of ode pq s of type pq (, c tp, tq, f) f t has c cycles of length pq, t p cycles of length p, t q cycles of length q and f fxed ponts n ts decomposton nto dsont cycles. A self-dual code C s doubly-even f all codewods of C have a weght dvsble by fou and sngly-even f thee s at least one codewod of weght conguent modulo 4. Rans n [1] poved that the mnmum dstance d of a bnay self-dual [ nkd,, ] code satsfes the followng bound: d 4 n/ 4 + 4, f n / ( mod 4 ), d 4 n/ 4 + 6, f n ( mod 4 ). Codes achevng ths bound ae called extemal. If n s a multple of 4, then a self-dual code meetng the bound must be doubly-even (see []). Moeove, fo any nonzeo weght w n such a code, the codewods of weght w fom a 5-desgn [3]. Ths s one eason why extemal codes of length 4m ae of patcula nteest. Unfotunately, only fo m = 1 and m = such codes ae known, namely the [4,1,8] extended Golay code and the [48,4,1] extended quadatc esdue code. Thus the exstence of no othe extemal code of length 4m s known. Fo n = 96, only the pmes, 3 and 5 may dvde the ode of the automophsm goup of the extemal code. We focus ou attenton on the case of an automophsm of ode 15. In [1] t s poved that a bnay doubly-even [96, 48, 0] self-dual code wth an automophsm of ode 15 does not exst. The queston of fndng a doubly-even self-dual [96, 48,16] code fst
3 Факултет по математика и информатика, том ХVІ С, 014 appeas n [4] whee also the fst such code was constucted. In ecent yeas such codes wth an automophsm of ode 3 ae constucted n [5] and [6]; fou codes ae known fom [7]; ten moe codes ae constucted n [8] and a code wth an automophsm of ode s constucted n [9]. We wll constuct many new doublyeven self-dual [96,48,16] codes wth an automophsm of ode 15 usng a method fo constuctng bnay self-dual codes nvaant unde the acton of a cyclc goup of ode pq fo odd pmes p q. The stuctue of the note s as follows. We begn wth shot descpton of the method n Secton (fo moe detals and poves we efe the eade to [1]). In Secton 3 we apply ths method to obtan codes wth n= 96, k = 48 and mnmum dstance 16 havng an automophsm of type 15-(6, 0,, 0).. Self-dual codes wth an automophsm of ode pq fo p < q odd pmes We consde the case = pq fo dffeent odd pmes p and q such that s a pmtve oot modulo p and modulo q. The gound feld s F. Then p 1 q 1 x 1 ( x 1) Q( xq ) ( xq ) ( x) (1 x)(1 x x = = )(1 + x+ + x ) Q( x), p q whee Q ( x ) s the -th cyclotomc polynomal. Moeove, both Q ( ) p x and Qq ( x ) ae educble ove F snce s a pmtve oot modulo p and modulo q as well. Fnally, f Q( x) = g3( x) gs( x) h1( x) h1( x) ht( x) ht ( x) s the factozaton of the -th cyclotomc polynomal nto educble factos ove F, then these factos have the same degee, φ() ( p 1)( q 1) namely =, whee φ s Eule's ph functon. s + t s + t Let the code C be a bnay self-dual codes possessng an automophsm of ode pq
4 Факултет по математика и информатика, том ХVІ С, 014 = Ω ΩΩ Ω Ω Ω Ω Ω σ 1 c c+ 1 c+ t 1 1, q c+ tq+ c+ tq+ tp c+ tq+ tp+ c+ tq+ tp+ f whee Ω = (( 1) + 1,, ) ae the pq cycles fo = 1,, c, Ω c+ = ( c + ( 1) q + 1,, c + q) cycles of length q fo = 1,, t q, Ω c+ t ( ( 1) 1,, ) q + = c + tqq + p + c + tqq + p cycles of length p fo = 1,, t p, and Ω c+ t ( ) q+ tp+ = c+ tq+ tp+ ae the fxed ponts fo = 1,, f. Let Fσ ( C) = { v C: vσ = v} and Eσ ( C) = { v C: wt( v Ω) 0 ( mod ), = 1,, c+ tq + tp}, whee v Ω s the estcton of v on Ω. Wth ths notaton we have the followng. Theoem 1 ([1]) The code C s a dect sum of the subcodes Fσ ( C) and Eσ ( C). n Let F be the n -dmensonal vecto space ove the bnay c+ tq+ tp+ f feld F, and let π : Fσ ( C) F be the poecton map,.e., ( π ( v)) = v fo some Ω and = 1,,, c+ tq + tp + f. Clealy, v Fσ ( C) ff v C and v s constant on each cycle of σ. Theoem ([1]) If C s a bnay self-dual code wth an automophsm σ of odd ode then Cπ = π ( Fσ( C)) s a bnay selfdual code of length c+ tq + tp + f. Consde the facto ng [ ]/ 1 R = F x x, whee x 1 s the pncpal deal n F [ x] geneated by x 1. Let x 1 = f ( x) f ( x) f ( x) be the factozaton of x 1 nto 0 1 s
5 Факултет по математика и информатика, том ХVІ С, 014 educble factos f ( x ) ove F whee f ( x) = x 1. Let 0 x 1 x 1 I = be the deal of R geneated by fo f ( x) f ( x) = 0,1,, s. By e ( x ) we denote the geneato dempotent of I ;.e., e ( x ) s the dentty of the two-sded deal I. Wth these notatons we have the followng esult (see [10]). Theoem 3 () R = I0 I1 Is. () I s a feld whch s somophc to the feld F deg ( f ( x )) fo l = 0,1,, s. () e ( x) e ( x ) = 0 fo. (v) s e ( x) = 1. = 0 Thee s a decomposton (see [11]) x 1 = g0( x) g1( x) gm( x) h1( x) h1( x) ht( x) ht ( x), whee s = m+ t and { g0, g1, gm, h1, h1,, ht, ht} = { f0, f1,, fs}. Futhemoe, h ( x ) s the ecpocal polynomal of h ( x ), h h fo = 1,, t and g ( x ) concdes wth ts ecpocal polynomal whee g0( x) = f0( x) = x 1. We denote the feld fo = 0,1,, m, H fo = 1,, t. x 1 h ( x) by x 1 g ( x) by x 1 H fo = 1,, t, and h ( x) G by
6 Факултет по математика и информатика, том ХVІ С, 014 σ ( C) be the shotened code of E ( C) Let E σ obtaned by emovng the last tq q + t p p+ f coodnates fom the codewods c c havng 0's thee. Next we defne a map ϕ : F R by c φ( v) = ( v ( x), v ( x),, v ( x)) R, whee 0 1 c v ( x) = v x and ( v0,, v, c 1) = v Ω. Clealy, ϕ ( C) s a lnea code ove the ng R of length c. Moeove, we have φ( C) = φ( C ) whee the dual code C ove F s taken unde the Eucldean nne poduct, and the dual code φ( C) c n R s taken wth espect to the Hemtan nne c 1 c 1 1 poduct: uv, = uv R, v = v( x ) = v( x ). In patcula, = 0 the code C s self-dual f and only f φ ( C) s self-dual ove R wth espect to the Hemtan nne poduct. Let Cϕ = ϕ( Eσ ( C) ). Snce Eσ ( C) s a bnay quas-cyclc code of length c and ndex c, C ϕ s a lnea code ove the ng R of m t length c. Moeove Cϕ = ( = 0M) ( 1( M = M )), M s a lnea code ove the feld G, = 1,, m, M s a lnea code ove H and M s a lnea code ove H, = 1,, t. Fo the dmensons we have dm E ( C) = dmc = σ φ = 0 whee s t ( p 1)( q 1) = ( p 1) dm M1+ ( q 1) dm M + ( dm M + (dm M + dm M )). s + t = = 3 1 Snce Eσ ( C) s a self-othogonal code, C ϕ s also selfothogonal ove the ng R wth espect to the Hemtan nne poduct. Ths means that M ae self-othogonal codes of length c
7 Факултет по математика и информатика, том ХVІ С, 014 ove G fo = 1,, m (wth espect to the Hemtan nne poduct) and, fo 1 t, we have M ( M ) wth espect to the Eucldean nne poduct. Ths foces dm M c / fo = 1,,, s and dm M + dm M c. It follows that c c ( p 1)( q 1) c c( pq 1) dm Eσ ( C) ( p 1) + ( q 1) + (( s ) + tc) =. s + t 3. Doubly-even self-dual [96, 48, 16] codes wth an automophsm of type 15 (6, 0,, 0) Let C be a doubly-even self-dual [96, 48, 16] code havng an automophsm of type 15 (6, 0,, 0). The weght dstbuton of such a code has the fom (see [7]) W( y) = 1 + ( α) y + ( α) y + ( α) y +, whee α s an ntege paamete. Codes wth α = 36918, 3733, 37608, 37884, 380, 38160, 3898, 38436, 38574, 3871, 38850, 38988, 3916, 3964, 3940, 39540, 39678, 39816, 39954, 4009, 4030, 40368, 40506, 4090, ae known fom [5]; α = 37500, 3754, 37584, ae fom [7]; the code n [4] has the weght enumeato coespondng to α = 377. Also the value α = s known fom [8] and α = 36864, 36876, 36888, 36900, 3691, 36936, 36948, 36960, 3697, ae fom [9]. We have that M 1 s a Hemtan self-othogonal [6,, ] code ove the feld G1 F 4, M s a Hemtan self-dual [6,3, d ] code ove G F 16, M s a lnea [6, k', d '] code ove H F 16 and M = ( M ) s ts dual code wth espect to the Eucldean nne poduct. Moeove, the code C has a geneato matx n the fom
8 Факултет по математика и информатика, том ХVІ С, π ( C ) π 1 ϕ ( M ) 0 1 ϕ ( M ) 0 (1) G =, 1 ϕ ( M ) 0 1 ϕ ( M1) ϕ ( D) ϕ ( I) genm 1 whee the matx geneates the dual code of M 1 ove G 1, D and I s the dentty matx ove the quatenay feld P 3. Fo the geneato matces of the codes M, M, M and C π we efe the eade to [1]. In shot 47 doubly-even self-othogonal [96,40,0] codes C96,40,1,, C96,40,47 wee constucted and we contnue to add the last 8 ows n G (comng fomϕ 1 1 ( M1) and ϕ ( D1 )) to obtan [96, 48,16] codes. Snce n evey pevous step we mpose the estcton that the mnmum dstance of the code s 0 we cannot gve full classfcaton. We have fou possble geneato matces fo M 1 : e1 0 e e1 0 e1 e1 e1 0 G1 =, G =, 0 e1 0 e e1 e1 xe1 x e1 0 e1 0 e e1 0 0 e1 e1 e1 G3 =, G4 =. 0 e1 0 e1 e1 e1 0 e1 e1 0 e1 e1 Table 1: The values of α and the numbe of [96, 48, 16] codes wth that α obtaned α # α # α # α # α # α # α #
9 Факултет по математика и информатика, том ХVІ С, Afte consdeng all pemutaton τ S6 of the columns of G1,, G4 and all ght cyclc shft n all 6 sx columns we found codes only when gen M1 = G3. The next theoem s a summay of the esults we have obtaned. Theoem 4 Thee exst at least bnay doubly-even [96, 48,16] self-dual codes wth an automophsm of type 15-(6, 0,, 0) of the obtaned codes have automophsm goups of ode 15; 763 have automophsm goups of ode 30, and 3 codes have goups of ode 45. The 19 values of the paamete α n the weght dstbuton W( y ) and the numbe of the constucted codes ae dsplayed n Table 1. Of all 19 only 8 values: 36876, 3691, 36936, 3697, 3733, 380, 38436, 3871 wee pevously know, so we obtan 11 new values of the paamete
10 Факултет по математика и информатика, том ХVІ С, 014 REFERENCES 1. Bouyukleva St., W. Wllems and N. Yankov, On the Automophsms of Ode 15 fo a Bnay Self-Dual [96, 48, 0] code. // axv: [cs.it], Rans E.M., Shadow bounds fo self-dual-codes. // IEEE Tans. Infom. Theoy, vol. 44, pp , Assmus E.F. and H.F. Mattson, New 5-desgns. // J. Combn. Theoy, vol. 6, pp , Fet W., A self-dual even (96, 48, 16) code. // IEEE Tans. Infom. Theoy, vol. 0, pp , Dontcheva R., Doubly-even self-dual code of length 96. // IEEE Tans. Inf. Theoy, vol. 48, no., pp , Yogova R. and A. Wassemann, Bnay self-dual codes wth automophsms of ode 3. // Des. Codes Cyptog., vol. 48, no., pp , S. T. Doughety, T. A. Gullve, and M. Haada, Extemal bnay self-dual codes. // IEEE Tans. Inf. Theoy, vol. 43, pp , Kaya A. and B. Yldz, Extenson theoems fo self-dual codes ove ngs and new bnay self-dual codes. // axv: [cs.it], Kaya A., B. Yldz, and I. Sap, New extemal bnay self-dual codes of length 68 fom quadatc esdue codes ove F + u F. + u F // Fnte Felds The Appl., vol. 9, pp , Huffman W.C., V. Pless, Fundamentals of Eo-Coectng Codes, Cambdge Unvesty Pess, Cambdge Lng S., P. Sole, On the algebac stuctue of quas-cyclc codes I: Fnte felds. // IEEE Tans. Infom. Theoy, vol. 47, pp ,
Set of square-integrable function 2 L : function space F
Set of squae-ntegable functon L : functon space F Motvaton: In ou pevous dscussons we have seen that fo fee patcles wave equatons (Helmholt o Schödnge) can be expessed n tems of egenvalue equatons. H E,
More informationGenerating Functions, Weighted and Non-Weighted Sums for Powers of Second-Order Recurrence Sequences
Geneatng Functons, Weghted and Non-Weghted Sums fo Powes of Second-Ode Recuence Sequences Pantelmon Stăncă Aubun Unvesty Montgomey, Depatment of Mathematcs Montgomey, AL 3614-403, USA e-mal: stanca@studel.aum.edu
More informationCorrespondence Analysis & Related Methods
Coespondence Analyss & Related Methods Ineta contbutons n weghted PCA PCA s a method of data vsualzaton whch epesents the tue postons of ponts n a map whch comes closest to all the ponts, closest n sense
More informationRanks of quotients, remainders and p-adic digits of matrices
axv:1401.6667v2 [math.nt] 31 Jan 2014 Ranks of quotents, emandes and p-adc dgts of matces Mustafa Elshekh Andy Novocn Mak Gesbecht Abstact Fo a pme p and a matx A Z n n, wte A as A = p(a quo p)+ (A em
More information8 Baire Category Theorem and Uniform Boundedness
8 Bae Categoy Theoem and Unfom Boundedness Pncple 8.1 Bae s Categoy Theoem Valdty of many esults n analyss depends on the completeness popety. Ths popety addesses the nadequacy of the system of atonal
More informationTHE ISOMORPHISM PROBLEM FOR CAYLEY GRAPHS ON THE GENERALIZED DICYCLIC GROUP
IJAMM 4:1 (016) 19-30 Mach 016 ISSN: 394-58 Avalale at http://scentfcadvances.co.n DOI: http://dx.do.og/10.1864/amml_710011617 THE ISOMORPHISM PROBEM FOR CAYEY RAPHS ON THE ENERAIZED DICYCIC ROUP Pedo
More informationarxiv: v1 [math.co] 1 Apr 2011
Weight enumeation of codes fom finite spaces Relinde Juius Octobe 23, 2018 axiv:1104.0172v1 [math.co] 1 Ap 2011 Abstact We study the genealized and extended weight enumeato of the - ay Simplex code and
More informationAPPLICATIONS OF SEMIGENERALIZED -CLOSED SETS
Intenatonal Jounal of Mathematcal Engneeng Scence ISSN : 22776982 Volume Issue 4 (Apl 202) http://www.mes.com/ https://stes.google.com/ste/mesounal/ APPLICATIONS OF SEMIGENERALIZED CLOSED SETS G.SHANMUGAM,
More informationP 365. r r r )...(1 365
SCIENCE WORLD JOURNAL VOL (NO4) 008 www.scecncewoldounal.og ISSN 597-64 SHORT COMMUNICATION ANALYSING THE APPROXIMATION MODEL TO BIRTHDAY PROBLEM *CHOJI, D.N. & DEME, A.C. Depatment of Mathematcs Unvesty
More informationEngineering Mechanics. Force resultants, Torques, Scalar Products, Equivalent Force systems
Engneeng echancs oce esultants, Toques, Scala oducts, Equvalent oce sstems Tata cgaw-hll Companes, 008 Resultant of Two oces foce: acton of one bod on anothe; chaacteed b ts pont of applcaton, magntude,
More informationDistinct 8-QAM+ Perfect Arrays Fanxin Zeng 1, a, Zhenyu Zhang 2,1, b, Linjie Qian 1, c
nd Intenatonal Confeence on Electcal Compute Engneeng and Electoncs (ICECEE 15) Dstnct 8-QAM+ Pefect Aays Fanxn Zeng 1 a Zhenyu Zhang 1 b Lnje Qan 1 c 1 Chongqng Key Laboatoy of Emegency Communcaton Chongqng
More informationRigid Bodies: Equivalent Systems of Forces
Engneeng Statcs, ENGR 2301 Chapte 3 Rgd Bodes: Equvalent Sstems of oces Intoducton Teatment of a bod as a sngle patcle s not alwas possble. In geneal, the se of the bod and the specfc ponts of applcaton
More informationChapter 3: Theory of Modular Arithmetic 38
Chapte 3: Theoy of Modula Aithmetic 38 Section D Chinese Remainde Theoem By the end of this section you will be able to pove the Chinese Remainde Theoem apply this theoem to solve simultaneous linea conguences
More informationScalars and Vectors Scalar
Scalas and ectos Scala A phscal quantt that s completel chaacteed b a eal numbe (o b ts numecal value) s called a scala. In othe wods a scala possesses onl a magntude. Mass denst volume tempeatue tme eneg
More informationGroupoid and Topological Quotient Group
lobal Jounal of Pue and Appled Mathematcs SSN 0973-768 Volume 3 Numbe 7 07 pp 373-39 Reseach nda Publcatons http://wwwpublcatoncom oupod and Topolocal Quotent oup Mohammad Qasm Manna Depatment of Mathematcs
More informationDilations and Commutant Lifting for Jointly Isometric OperatorsA Geometric Approach
jounal of functonal analyss 140, 300311 (1996) atcle no. 0109 Dlatons and Commutant Lftng fo Jontly Isometc OpeatosA Geometc Appoach K. R. M. Attele and A. R. Lubn Depatment of Mathematcs, Illnos Insttute
More informationV. Principles of Irreversible Thermodynamics. s = S - S 0 (7.3) s = = - g i, k. "Flux": = da i. "Force": = -Â g a ik k = X i. Â J i X i (7.
Themodynamcs and Knetcs of Solds 71 V. Pncples of Ievesble Themodynamcs 5. Onsage s Teatment s = S - S 0 = s( a 1, a 2,...) a n = A g - A n (7.6) Equlbum themodynamcs detemnes the paametes of an equlbum
More informationMachine Learning. Spectral Clustering. Lecture 23, April 14, Reading: Eric Xing 1
Machne Leanng -7/5 7/5-78, 78, Spng 8 Spectal Clusteng Ec Xng Lectue 3, pl 4, 8 Readng: Ec Xng Data Clusteng wo dffeent ctea Compactness, e.g., k-means, mxtue models Connectvty, e.g., spectal clusteng
More informationUNIT10 PLANE OF REGRESSION
UIT0 PLAE OF REGRESSIO Plane of Regesson Stuctue 0. Intoducton Ojectves 0. Yule s otaton 0. Plane of Regesson fo thee Vaales 0.4 Popetes of Resduals 0.5 Vaance of the Resduals 0.6 Summay 0.7 Solutons /
More informationChapter I Matrices, Vectors, & Vector Calculus 1-1, 1-9, 1-10, 1-11, 1-17, 1-18, 1-25, 1-27, 1-36, 1-37, 1-41.
Chapte I Matces, Vectos, & Vecto Calculus -, -9, -0, -, -7, -8, -5, -7, -36, -37, -4. . Concept of a Scala Consde the aa of patcles shown n the fgue. he mass of the patcle at (,) can be epessed as. M (,
More informationarxiv: v2 [cs.it] 11 Jul 2014
A faly of optal locally ecoveable codes Itzhak Tao, Mebe, IEEE, and Alexande Bag, Fellow, IEEE axv:1311.3284v2 [cs.it] 11 Jul 2014 Abstact A code ove a fnte alphabet s called locally ecoveable (LRC) f
More informationONE-POINT CODES USING PLACES OF HIGHER DEGREE
ONE-POINT CODES USING PLACES OF HIGHER DEGREE GRETCHEN L. MATTHEWS AND TODD W. MICHEL DEPARTMENT OF MATHEMATICAL SCIENCES CLEMSON UNIVERSITY CLEMSON, SC 29634-0975 U.S.A. E-MAIL: GMATTHE@CLEMSON.EDU, TMICHEL@CLEMSON.EDU
More informationA. Thicknesses and Densities
10 Lab0 The Eath s Shells A. Thcknesses and Denstes Any theoy of the nteo of the Eath must be consstent wth the fact that ts aggegate densty s 5.5 g/cm (ecall we calculated ths densty last tme). In othe
More information9.1 The multiplicative group of a finite field. Theorem 9.1. The multiplicative group F of a finite field is cyclic.
Chapte 9 Pimitive Roots 9.1 The multiplicative goup of a finite fld Theoem 9.1. The multiplicative goup F of a finite fld is cyclic. Remak: In paticula, if p is a pime then (Z/p) is cyclic. In fact, this
More information24-2: Electric Potential Energy. 24-1: What is physics
D. Iyad SAADEDDIN Chapte 4: Electc Potental Electc potental Enegy and Electc potental Calculatng the E-potental fom E-feld fo dffeent chage dstbutons Calculatng the E-feld fom E-potental Potental of a
More information4 SingularValue Decomposition (SVD)
/6/00 Z:\ jeh\self\boo Kannan\Jan-5-00\4 SVD 4 SngulaValue Decomposton (SVD) Chapte 4 Pat SVD he sngula value decomposton of a matx s the factozaton of nto the poduct of thee matces = UDV whee the columns
More informationNew problems in universal algebraic geometry illustrated by boolean equations
New poblems in univesal algebaic geomety illustated by boolean equations axiv:1611.00152v2 [math.ra] 25 Nov 2016 Atem N. Shevlyakov Novembe 28, 2016 Abstact We discuss new poblems in univesal algebaic
More informationHamiltonian multivector fields and Poisson forms in multisymplectic field theory
JOURNAL OF MATHEMATICAL PHYSICS 46, 12005 Hamltonan multvecto felds and Posson foms n multsymplectc feld theoy Mchael Foge a Depatamento de Matemátca Aplcada, Insttuto de Matemátca e Estatístca, Unvesdade
More informationGENERALIZATION OF AN IDENTITY INVOLVING THE GENERALIZED FIBONACCI NUMBERS AND ITS APPLICATIONS
#A39 INTEGERS 9 (009), 497-513 GENERALIZATION OF AN IDENTITY INVOLVING THE GENERALIZED FIBONACCI NUMBERS AND ITS APPLICATIONS Mohaad Faokh D. G. Depatent of Matheatcs, Fedows Unvesty of Mashhad, Mashhad,
More informationPhysics 11b Lecture #2. Electric Field Electric Flux Gauss s Law
Physcs 11b Lectue # Electc Feld Electc Flux Gauss s Law What We Dd Last Tme Electc chage = How object esponds to electc foce Comes n postve and negatve flavos Conseved Electc foce Coulomb s Law F Same
More informationMachine Learning 4771
Machne Leanng 4771 Instucto: Tony Jebaa Topc 6 Revew: Suppot Vecto Machnes Pmal & Dual Soluton Non-sepaable SVMs Kenels SVM Demo Revew: SVM Suppot vecto machnes ae (n the smplest case) lnea classfes that
More informationCapítulo. Three Dimensions
Capítulo Knematcs of Rgd Bodes n Thee Dmensons Mecánca Contents ntoducton Rgd Bod Angula Momentum n Thee Dmensons Pncple of mpulse and Momentum Knetc Eneg Sample Poblem 8. Sample Poblem 8. Moton of a Rgd
More informationLinks in edge-colored graphs
Lnks n edge-coloed gaphs J.M. Becu, M. Dah, Y. Manoussaks, G. Mendy LRI, Bât. 490, Unvesté Pas-Sud 11, 91405 Osay Cedex, Fance Astact A gaph s k-lnked (k-edge-lnked), k 1, f fo each k pas of vetces x 1,
More informationA STUDY OF HAMMING CODES AS ERROR CORRECTING CODES
AGU Intenational Jounal of Science and Technology A STUDY OF HAMMING CODES AS ERROR CORRECTING CODES Ritu Ahuja Depatment of Mathematics Khalsa College fo Women, Civil Lines, Ludhiana-141001, Punjab, (India)
More informationEnergy in Closed Systems
Enegy n Closed Systems Anamta Palt palt.anamta@gmal.com Abstact The wtng ndcates a beakdown of the classcal laws. We consde consevaton of enegy wth a many body system n elaton to the nvese squae law and
More informationChapter Fifiteen. Surfaces Revisited
Chapte Ffteen ufaces Revsted 15.1 Vecto Descpton of ufaces We look now at the vey specal case of functons : D R 3, whee D R s a nce subset of the plane. We suppose s a nce functon. As the pont ( s, t)
More informationKhintchine-Type Inequalities and Their Applications in Optimization
Khntchne-Type Inequaltes and The Applcatons n Optmzaton Anthony Man-Cho So Depatment of Systems Engneeng & Engneeng Management The Chnese Unvesty of Hong Kong ISDS-Kolloquum Unvestaet Wen 29 June 2009
More informationThermodynamics of solids 4. Statistical thermodynamics and the 3 rd law. Kwangheon Park Kyung Hee University Department of Nuclear Engineering
Themodynamcs of solds 4. Statstcal themodynamcs and the 3 d law Kwangheon Pak Kyung Hee Unvesty Depatment of Nuclea Engneeng 4.1. Intoducton to statstcal themodynamcs Classcal themodynamcs Statstcal themodynamcs
More informationCOMPLEMENTARY ENERGY METHOD FOR CURVED COMPOSITE BEAMS
ultscence - XXX. mcocd Intenatonal ultdscplnay Scentfc Confeence Unvesty of skolc Hungay - pl 06 ISBN 978-963-358-3- COPLEENTRY ENERGY ETHOD FOR CURVED COPOSITE BES Ákos József Lengyel István Ecsed ssstant
More informationEnumerating permutation polynomials
Enumeating pemutation polynomials Theodoulos Gaefalakis a,1, Giogos Kapetanakis a,, a Depatment of Mathematics and Applied Mathematics, Univesity of Cete, 70013 Heaklion, Geece Abstact We conside thoblem
More informationIf there are k binding constraints at x then re-label these constraints so that they are the first k constraints.
Mathematcal Foundatons -1- Constaned Optmzaton Constaned Optmzaton Ma{ f ( ) X} whee X {, h ( ), 1,, m} Necessay condtons fo to be a soluton to ths mamzaton poblem Mathematcally, f ag Ma{ f ( ) X}, then
More informationSummer Workshop on the Reaction Theory Exercise sheet 8. Classwork
Joned Physcs Analyss Cente Summe Wokshop on the Reacton Theoy Execse sheet 8 Vncent Matheu Contact: http://www.ndana.edu/~sst/ndex.html June June To be dscussed on Tuesday of Week-II. Classwok. Deve all
More informationMultistage Median Ranked Set Sampling for Estimating the Population Median
Jounal of Mathematcs and Statstcs 3 (: 58-64 007 ISSN 549-3644 007 Scence Publcatons Multstage Medan Ranked Set Samplng fo Estmatng the Populaton Medan Abdul Azz Jeman Ame Al-Oma and Kamaulzaman Ibahm
More informationPart V: Velocity and Acceleration Analysis of Mechanisms
Pat V: Velocty an Acceleaton Analyss of Mechansms Ths secton wll evew the most common an cuently pactce methos fo completng the knematcs analyss of mechansms; escbng moton though velocty an acceleaton.
More informationA. Proofs for learning guarantees
Leanng Theoy and Algoths fo Revenue Optzaton n Second-Pce Auctons wth Reseve A. Poofs fo leanng guaantees A.. Revenue foula The sple expesson of the expected evenue (2) can be obtaned as follows: E b Revenue(,
More informationPHYS 705: Classical Mechanics. Derivation of Lagrange Equations from D Alembert s Principle
1 PHYS 705: Classcal Mechancs Devaton of Lagange Equatons fom D Alembet s Pncple 2 D Alembet s Pncple Followng a smla agument fo the vtual dsplacement to be consstent wth constants,.e, (no vtual wok fo
More informationOptimization Methods: Linear Programming- Revised Simplex Method. Module 3 Lecture Notes 5. Revised Simplex Method, Duality and Sensitivity analysis
Optmzaton Meods: Lnea Pogammng- Revsed Smple Meod Module Lectue Notes Revsed Smple Meod, Dualty and Senstvty analyss Intoducton In e pevous class, e smple meod was dscussed whee e smple tableau at each
More information19 The Born-Oppenheimer Approximation
9 The Bon-Oppenheme Appoxmaton The full nonelatvstc Hamltonan fo a molecule s gven by (n a.u.) Ĥ = A M A A A, Z A + A + >j j (883) Lets ewte the Hamltonan to emphasze the goal as Ĥ = + A A A, >j j M A
More informationEfficiency of the principal component Liu-type estimator in logistic
Effcency of the pncpal component Lu-type estmato n logstc egesson model Jbo Wu and Yasn Asa 2 School of Mathematcs and Fnance, Chongqng Unvesty of Ats and Scences, Chongqng, Chna 2 Depatment of Mathematcs-Compute
More informationA Repair Framework for Scalar MDS Codes
axv:32235v [csit] 7 Dec 203 A Repa Famewo fo Scala MDS Codes Katheyan Shanmugam Student Membe, IEEE, Dmts S apalopoulos Student Membe, IEEE, Alexandos G Dmas Membe, IEEE, and Guseppe Cae Fellow, IEEE Depatment
More informationIMA Preprint Series # 2202
FRIENDY EQUIIBRIUM INTS IN EXTENSIVE GMES WITH CMETE INFRMTIN By Ezo Mch IM epnt Sees # My 8 INSTITUTE FR MTHEMTICS ND ITS ICTINS UNIVERSITY F MINNEST nd Hll 7 Chuch Steet S.E. Mnnepols Mnnesot 5555 6
More informationOn the Distribution of the Product and Ratio of Independent Central and Doubly Non-central Generalized Gamma Ratio random variables
On the Dstbuton of the Poduct Rato of Independent Cental Doubly Non-cental Genealzed Gamma Rato om vaables Calos A. Coelho João T. Mexa Abstact Usng a decomposton of the chaactestc functon of the logathm
More informationA Tutorial on Low Density Parity-Check Codes
A Tutoal on Low Densty Paty-Check Codes Tuan Ta The Unvesty of Texas at Austn Abstact Low densty paty-check codes ae one of the hottest topcs n codng theoy nowadays. Equpped wth vey fast encodng and decodng
More informationBayesian Assessment of Availabilities and Unavailabilities of Multistate Monotone Systems
Dept. of Math. Unvesty of Oslo Statstcal Reseach Repot No 3 ISSN 0806 3842 June 2010 Bayesan Assessment of Avalabltes and Unavalabltes of Multstate Monotone Systems Bent Natvg Jøund Gåsemy Tond Retan June
More informationDIFFERENTIAL FORMS BRIAN OSSERMAN
DIFFERENTIAL FORMS BRIAN OSSERMAN Dfferentals are an mportant topc n algebrac geometry, allowng the use of some classcal geometrc arguments n the context of varetes over any feld. We wll use them to defne
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationINTRODUCTION. consider the statements : I there exists x X. f x, such that. II there exists y Y. such that g y
INRODUCION hs dssetaton s the eadng of efeences [1], [] and [3]. Faas lemma s one of the theoems of the altenatve. hese theoems chaacteze the optmalt condtons of seveal mnmzaton poblems. It s nown that
More informationDYNAMICS VECTOR MECHANICS FOR ENGINEERS: Kinematics of Rigid Bodies in Three Dimensions. Seventh Edition CHAPTER
Edton CAPTER 8 VECTOR MECANCS FOR ENGNEERS: DYNAMCS Fednand P. Bee E. Russell Johnston, J. Lectue Notes: J. Walt Ole Teas Tech Unvest Knematcs of Rgd Bodes n Thee Dmensons 003 The McGaw-ll Companes, nc.
More informationON THE INVERSE SIGNED TOTAL DOMINATION NUMBER IN GRAPHS. D.A. Mojdeh and B. Samadi
Opuscula Math. 37, no. 3 (017), 447 456 http://dx.doi.og/10.7494/opmath.017.37.3.447 Opuscula Mathematica ON THE INVERSE SIGNED TOTAL DOMINATION NUMBER IN GRAPHS D.A. Mojdeh and B. Samadi Communicated
More informationMechanics Physics 151
Mechancs Physcs 151 Lectue 18 Hamltonan Equatons of Moton (Chapte 8) What s Ahead We ae statng Hamltonan fomalsm Hamltonan equaton Today and 11/6 Canoncal tansfomaton 1/3, 1/5, 1/10 Close lnk to non-elatvstc
More informationA BANDWIDTH THEOREM FOR APPROXIMATE DECOMPOSITIONS
A BANDWIDTH THEOREM FOR APPROXIMATE DECOMPOSITIONS PADRAIG CONDON, JAEHOON KIM, DANIELA KÜHN AND DERYK OSTHUS Abstact. We povde a degee condton on a egula n-vetex gaph G whch ensues the exstence of a nea
More informationPhysics Exam II Chapters 25-29
Physcs 114 1 Exam II Chaptes 5-9 Answe 8 of the followng 9 questons o poblems. Each one s weghted equally. Clealy mak on you blue book whch numbe you do not want gaded. If you ae not sue whch one you do
More informationE. JAVIER ELIZONDO, KAZUHIKO KURANO, AND KEI-ICHI WATANABE s a Noethean ng fo any Wel dvso D on X (e.g., Elzondo and Snvas [ES0]). uthemoe, H 0 (X O X
THE TOTAL COORDINATE RING O A NORAL PROJECTIVE VARIETY E. JAVIER ELIZONDO, KAZUHIKO KURANO, AND KEI-ICHI WATANABE. Intoducton The total coodnate ng TC(X) of a vaety s a genealzaton of the ng ntoduced and
More informationgravity r2,1 r2 r1 by m 2,1
Gavtaton Many of the foundatons of classcal echancs wee fst dscoveed when phlosophes (ealy scentsts and atheatcans) ted to explan the oton of planets and stas. Newton s ost faous fo unfyng the oton of
More informationThe Forming Theory and the NC Machining for The Rotary Burs with the Spectral Edge Distribution
oden Appled Scence The Fomn Theoy and the NC achnn fo The Rotay us wth the Spectal Ede Dstbuton Huan Lu Depatment of echancal Enneen, Zhejan Unvesty of Scence and Technoloy Hanzhou, c.y. chan, 310023,
More informationq-bernstein polynomials and Bézier curves
Jounal of Computatonal and Appled Mathematcs 151 (2003) 1-12 q-bensten polynomals and Béze cuves Hall Ouç a, and Geoge M. Phllps b a Depatment of Mathematcs, Dokuz Eylül Unvesty Fen Edebyat Fakültes, Tınaztepe
More informationIntegral Vector Operations and Related Theorems Applications in Mechanics and E&M
Dola Bagayoko (0) Integal Vecto Opeatons and elated Theoems Applcatons n Mechancs and E&M Ι Basc Defnton Please efe to you calculus evewed below. Ι, ΙΙ, andιιι notes and textbooks fo detals on the concepts
More informationPhysics 202, Lecture 2. Announcements
Physcs 202, Lectue 2 Today s Topcs Announcements Electc Felds Moe on the Electc Foce (Coulomb s Law The Electc Feld Moton of Chaged Patcles n an Electc Feld Announcements Homewok Assgnment #1: WebAssgn
More informationON THE FRESNEL SINE INTEGRAL AND THE CONVOLUTION
IJMMS 3:37, 37 333 PII. S16117131151 http://jmms.hndaw.com Hndaw Publshng Cop. ON THE FRESNEL SINE INTEGRAL AND THE CONVOLUTION ADEM KILIÇMAN Receved 19 Novembe and n evsed fom 7 Mach 3 The Fesnel sne
More informationChapter 8. Linear Momentum, Impulse, and Collisions
Chapte 8 Lnea oentu, Ipulse, and Collsons 8. Lnea oentu and Ipulse The lnea oentu p of a patcle of ass ovng wth velocty v s defned as: p " v ote that p s a vecto that ponts n the sae decton as the velocty
More informationPhysics 2A Chapter 11 - Universal Gravitation Fall 2017
Physcs A Chapte - Unvesal Gavtaton Fall 07 hese notes ae ve pages. A quck summay: he text boxes n the notes contan the esults that wll compse the toolbox o Chapte. hee ae thee sectons: the law o gavtaton,
More informationPattern Analyses (EOF Analysis) Introduction Definition of EOFs Estimation of EOFs Inference Rotated EOFs
Patten Analyses (EOF Analyss) Intoducton Defnton of EOFs Estmaton of EOFs Infeence Rotated EOFs . Patten Analyses Intoducton: What s t about? Patten analyses ae technques used to dentfy pattens of the
More informationRemember: When an object falls due to gravity its potential energy decreases.
Chapte 5: lectc Potental As mentoned seveal tmes dung the uate Newton s law o gavty and Coulomb s law ae dentcal n the mathematcal om. So, most thngs that ae tue o gavty ae also tue o electostatcs! Hee
More informationAsymptotic Waves for a Non Linear System
Int Jounal of Math Analyss, Vol 3, 9, no 8, 359-367 Asymptotc Waves fo a Non Lnea System Hamlaou Abdelhamd Dépatement de Mathématques, Faculté des Scences Unvesté Bad Mokhta BP,Annaba, Algea hamdhamlaou@yahoof
More informationTest 1 phy What mass of a material with density ρ is required to make a hollow spherical shell having inner radius r i and outer radius r o?
Test 1 phy 0 1. a) What s the pupose of measuement? b) Wte all fou condtons, whch must be satsfed by a scala poduct. (Use dffeent symbols to dstngush opeatons on ectos fom opeatons on numbes.) c) What
More informationAn Introduction to Morita Theory
An Introducton to Morta Theory Matt Booth October 2015 Nov. 2017: made a few revsons. Thanks to Nng Shan for catchng a typo. My man reference for these notes was Chapter II of Bass s book Algebrac K-Theory
More informationVanishing lines in generalized Adams spectral sequences are generic
ISSN 364-0380 (on line) 465-3060 (pinted) 55 Geomety & Topology Volume 3 (999) 55 65 Published: 2 July 999 G G G G T T T G T T T G T G T GG TT G G G G GG T T T TT Vanishing lines in genealized Adams spectal
More informationPhysics 1501 Lecture 19
Physcs 1501 ectue 19 Physcs 1501: ectue 19 Today s Agenda Announceents HW#7: due Oct. 1 Mdte 1: aveage 45 % Topcs otatonal Kneatcs otatonal Enegy Moents of Ineta Physcs 1501: ectue 19, Pg 1 Suay (wth copason
More informationALL QUESTIONS ARE WORTH 20 POINTS. WORK OUT FIVE PROBLEMS.
GNRAL PHYSICS PH -3A (D. S. Mov) Test (/3/) key STUDNT NAM: STUDNT d #: -------------------------------------------------------------------------------------------------------------------------------------------
More informationLecture 5 Single factor design and analysis
Lectue 5 Sngle fcto desgn nd nlss Completel ndomzed desgn (CRD Completel ndomzed desgn In the desgn of expements, completel ndomzed desgns e fo studng the effects of one pm fcto wthout the need to tke
More information6.6 The Marquardt Algorithm
6.6 The Mqudt Algothm lmttons of the gdent nd Tylo expnson methods ecstng the Tylo expnson n tems of ch-sque devtves ecstng the gdent sech nto n tetve mtx fomlsm Mqudt's lgothm utomtclly combnes the gdent
More informationISSN: ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 3, Issue 1, July 2013
ISSN: 2277-375 Constructon of Trend Free Run Orders for Orthogonal rrays Usng Codes bstract: Sometmes when the expermental runs are carred out n a tme order sequence, the response can depend on the run
More informationSolution to HW 3, Ma 1a Fall 2016
Solution to HW 3, Ma a Fall 206 Section 2. Execise 2: Let C be a subset of the eal numbes consisting of those eal numbes x having the popety that evey digit in the decimal expansion of x is, 3, 5, o 7.
More informationPHY126 Summer Session I, 2008
PHY6 Summe Sesson I, 8 Most of nfomaton s avalable at: http://nngoup.phscs.sunsb.edu/~chak/phy6-8 ncludng the sllabus and lectue sldes. Read sllabus and watch fo mpotant announcements. Homewok assgnment
More informationChapter 5 Linear Equations: Basic Theory and Practice
Chapte 5 inea Equations: Basic Theoy and actice In this chapte and the next, we ae inteested in the linea algebaic equation AX = b, (5-1) whee A is an m n matix, X is an n 1 vecto to be solved fo, and
More informationGoodness-of-fit for composite hypotheses.
Section 11 Goodness-of-fit fo composite hypotheses. Example. Let us conside a Matlab example. Let us geneate 50 obsevations fom N(1, 2): X=nomnd(1,2,50,1); Then, unning a chi-squaed goodness-of-fit test
More informationCOLLAPSING WALLS THEOREM
COLLAPSING WALLS THEOREM IGOR PAK AND ROM PINCHASI Abstact. Let P R 3 be a pyamid with the base a convex polygon Q. We show that when othe faces ae collapsed (otated aound the edges onto the plane spanned
More informationCSJM University Class: B.Sc.-II Sub:Physics Paper-II Title: Electromagnetics Unit-1: Electrostatics Lecture: 1 to 4
CSJM Unvesty Class: B.Sc.-II Sub:Physcs Pape-II Ttle: Electomagnetcs Unt-: Electostatcs Lectue: to 4 Electostatcs: It deals the study of behavo of statc o statonay Chages. Electc Chage: It s popety by
More information(n 1)n(n + 1)(n + 2) + 1 = (n 1)(n + 2)n(n + 1) + 1 = ( (n 2 + n 1) 1 )( (n 2 + n 1) + 1 ) + 1 = (n 2 + n 1) 2.
Paabola Volume 5, Issue (017) Solutions 151 1540 Q151 Take any fou consecutive whole numbes, multiply them togethe and add 1. Make a conjectue and pove it! The esulting numbe can, fo instance, be expessed
More informationEvent Shape Update. T. Doyle S. Hanlon I. Skillicorn. A. Everett A. Savin. Event Shapes, A. Everett, U. Wisconsin ZEUS Meeting, October 15,
Event Shape Update A. Eveett A. Savn T. Doyle S. Hanlon I. Skllcon Event Shapes, A. Eveett, U. Wsconsn ZEUS Meetng, Octobe 15, 2003-1 Outlne Pogess of Event Shapes n DIS Smla to publshed pape: Powe Coecton
More informationDynamics of Rigid Bodies
Dynamcs of Rgd Bodes A gd body s one n whch the dstances between consttuent patcles s constant thoughout the moton of the body,.e. t keeps ts shape. Thee ae two knds of gd body moton: 1. Tanslatonal Rectlnea
More informationAsymptotic Solutions of the Kinetic Boltzmann Equation and Multicomponent Non-Equilibrium Gas Dynamics
Jounal of Appled Mathematcs and Physcs 6 4 687-697 Publshed Onlne August 6 n ScRes http://wwwscpog/jounal/jamp http://dxdoog/436/jamp64877 Asymptotc Solutons of the Knetc Boltzmann Equaton and Multcomponent
More informationThe Backpropagation Algorithm
The Backpopagaton Algothm Achtectue of Feedfowad Netwok Sgmodal Thehold Functon Contuctng an Obectve Functon Tanng a one-laye netwok by teepet decent Tanng a two-laye netwok by teepet decent Copyght Robet
More informationtitrrvers:rtt t>1 NO~~H CAROLINA
titvers:tt t>1 NO~~H CAROLINA Depatment of statistics Chapel Hill, N. C. ON A BOUN.D USEFUL IN THE THEORY OF FACTORIAL DESIGNS AND ERROR CORRECTING CODES by R. C. Bose and J. N. Sivastava Apil 1963 Gant
More informationSL n (F ) Equals its Own Derived Group
Internatonal Journal of Algebra, Vol. 2, 2008, no. 12, 585-594 SL n (F ) Equals ts Own Derved Group Jorge Macel BMCC-The Cty Unversty of New York, CUNY 199 Chambers street, New York, NY 10007, USA macel@cms.nyu.edu
More informationDirichlet Mixture Priors: Inference and Adjustment
Dchlet Mxtue Pos: Infeence and Adustment Xugang Ye (Wokng wth Stephen Altschul and Y Kuo Yu) Natonal Cante fo Botechnology Infomaton Motvaton Real-wold obects Independent obsevatons Categocal data () (2)
More informationChapter 13 - Universal Gravitation
Chapte 3 - Unesal Gataton In Chapte 5 we studed Newton s thee laws of moton. In addton to these laws, Newton fomulated the law of unesal gataton. Ths law states that two masses ae attacted by a foce gen
More informationRotary motion
ectue 8 RTARY TN F THE RGD BDY Notes: ectue 8 - Rgd bod Rgd bod: j const numbe of degees of feedom 6 3 tanslatonal + 3 ota motons m j m j Constants educe numbe of degees of feedom non-fee object: 6-p
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
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 information