In 1991 Fermat s Last Theorem Has Been Proved
|
|
- Stewart Newman
- 6 years ago
- Views:
Transcription
1 I 99 Frmat s Last Thorm Has B Provd Chu-Xua Jag P.O.Box 94Bg 00854Cha Jcxua00@s.com;cxxxx@6.com bstract I 67 Frmat wrot: It s mpossbl to sparat a cub to two cubs or a bquadrat to two bquadrats or gral ay powr hghr tha th scod to powrs of lk dgr: I hav dscovrd a truly marvlous proof whch ths marg s too small to cota. Ths mas: x + y = z ( > ) has o tgr solutos all dffrt from 0(.. t has oly th trval soluto whr o of th tgrs s qual to 0). It has b calld Frmat s last thorm (FLT). It suffcs to prov FLT for xpot 4. ad vry prm xpot P. Frmat provd FLT for xpot 4. Eulr provd FLT for xpot.o Octobr 599 w provd Frmat last thorm[]. I ths papr usg th complx hyprbolc fuctos w prov FLT for xpots P ad P whr P s a odd prm. Th proof of FLT must b drct. But drct proof of FLT s dsblvg. I 974 Jag foud out Eulr formula of th cyclotomc ral umbrs th cyclotomc flds xp tj = J () = = whr J dots a th root of uty J = s a odd umbr t ar th ral umbrs. s calld th complx hyprbolc fuctos of ordr wth varabls [-7]. ( ) B ( ) = + ( ) cos θ + ( ) () 000 mathmatcs subct classfcato D4. Ky words complx hyprbolc Eulr proof.
2 whr = ; = t α α α B = tα ( ) cos α α = + α α θ = ( ) tα ( ) s α = () may b wrtt th matrx form + B = 0 () 0 0 ( ) cos s s ( ) = cos s s ( ) ( ) ( ) cos s s B cosθ B sθ xp B sθ (4) whr ( ) / s a v umbr. From (4) w hav ts vrs trasformato ( ) cos cos cos B cosθ B sθ ( ) = 0 s s s xp( B )s( θ ) ( ) ( ) ( ) 0 s s s From (5) w hav (5) = + ( ) cos = B cos θ + = = s ( ) ( ) s (6) B + θ = + = I () ad (6) t ad hav th sam formulas. (4) ad (5) ar th most crtcal formulas of proofs for FLT. Usg (4) ad (5) 99 Jag vtd that vry factor of xpot has th Frmat quato ad provd FLT [-7] ubsttutg (4) to (5) w prov (5).
3 ( ) cos cos cos B cosθ B sθ ( ) = 0 s s s xp( B )s( θ ) ( ) ( ) ( ) 0 s s s 0 0 ( ) cos s s B cosθ ( ) B sθ cos s s xp( B )s( θ ) ( ) ( ) ( ) cos s s B cosθ B sθ = xp( B )s( θ ) cosθ B B sθ = (7) xp( B )s( θ ) whr + (cos ) = (s ) =. From () w hav From (6) w hav xp( + B ) =. (8)
4 ( ) ( ) ( ) ( ) xp( + B ) = = ( ) ( ) (9) whr ( ) = [7]. t From (8) ad (9) w hav th crculat dtrmat xp( + B ) = = (0) If 0 whr = th (0) has ftly may ratoal solutos. ssum 0 0 = 0 whr = 4. = 0 ar dtrmat quatos wth varabls. From (6) w hav B = + = + + ( ) cos. () From (0) ad () w hav th Frmat quato xp( + B) = ( + ) Π ( + + ( ) cos ) = + = () Exampl[]. Lt = 5. From () w hav = ( t + t ) + ( t + t ) + ( t + t ) + ( t + t ) + ( t + t ) + ( t + t ) + ( t + t ) B = ( t + t4)cos + ( t + t)cos ( t + t)cos + ( t4 + t)cos ( t5 + t0)cos + ( t6 + t9)cos ( t7 + t8)cos B = ( t + t4)cos + ( t + t)cos + ( t + t)cos + ( t4 + t)cos ( t5 + t )cos + ( t06 + t9)cos + ( t7 + t8)cos B = ( t + t4)cos + ( t + t)cos ( t + t)cos + ( t4 + t)cos ( t5 + t0)cos + ( t6 + t9)cos ( t7 + t8)cos B4 = ( t + t4)cos + ( t + t)cos + ( t + t)cos + ( t4 + t)cos ( t5 + t )cos + ( 06 t + t9)cos + ( t7 + t8)cos
5 B5 = ( t + t4)cos + ( t + t)cos ( t + t)cos + ( t4 + t)cos ( t5 + t0)cos + ( t6 + t9)cos ( t7 + t8)cos B6 = ( t + t4)cos + ( t + t)cos + ( t + t)cos + ( t4 + t)cos ( t5 + t )cos + ( t06 + t9)cos + ( t7 + t8)cos B7 = ( t + t4)cos + ( t + t)cos ( t + t)cos + ( t4 + t)cos ( t5 + t0)cos + ( t6 + t9)cos ( t7 + t8)cos () + B = 0 + B + B = 5( t + t ) Form () w hav th Frmat quato From () w hav From () w hav (4) xp( + B ) = + = ( ) + ( ) = xp( B B ) [xp( t t )] From (5) ad (6) w hav th Frmat quato = (5) xp( + B + B ) = +. (6) xp( + B + B ) = + = [xp( t + t )]. (7) Eulr provd that (4) has o ratoal solutos for xpot [8]. Thrfor w prov that (7) has o ratoal solutos for xpot 5[]. Thorm. [-7]. Lt = Pwhr P > s odd prm. From () w hav th Frmat s quato From () w hav From () w hav P P P P P. (8) xp( + B ) = + = ( ) + ( ) = P P P P. (9) xp( + B ) = [xp( t + t )] P P P. (0) xp( + B ) = + From (9) ad (0) w hav th Frmat quato 5
6 P P P P P P. () xp( + B ) = + = [xp( t + t )] Eulr provd that (8) has o ratoal solutos for xpot [8]. Thrfor w prov that () has o ratoal solutos for P > [ -7]. Not. Wls had ot provd Frmat last thorm[9-] Rfrcs [] Jag C-X Frmat last thorm had b provd Pottal cc ( Chs).7-0 (99) Prprts ( Eglsh) Dcmbr (99). [] Jag C-X Frmat last thorm had b provd by Frmat mor tha 00 yars ago Pottal cc ( Chs) 6.8-0(99). [] Jag C-X O th factorzato thorm of crculat dtrmat lgbras Groups ad Gomtrs. 7-77(994) MR. 96a: 0 [4] Jag C-X Frmat last thorm was provd 99 Prprts (99). I: Fudamtal op problms scc at th d of th mllum T.Gll K. Lu ad E. Trll (ds). Hadroc Prss 999 P [5] Jag C-X O th Frmat-atll thorm lgbras Groups ad Gomtrs (998) [6] Jag C-X Complx hyprbolc fuctos ad Frmat s last thorm Hadroc Joural upplmt (000). [7] Jag C-X Foudatos of atll Isoumbr Thory wth applcatos to w cryptograms Frmat s thorm ad Goldbach s Coctur. Itr cad. Prss. 00. MR004c: [8] Rbbom P Frmat last thorm for amatur prgr-vrlag (999). [9] Wls Modular llptc curvs ad Frmat last thorm. Math 4()44-55(995). [0] ZhvolovY Frmat last thorm ad mstaks of drw Wls [] ZhvolovY Frmat last thorm ad Kth Rbt mstaks
Aotomorphic Functions And Fermat s Last Theorem(4)
otomorphc Fuctos d Frmat s Last Thorm(4) Chu-Xua Jag P. O. Box 94 Bg 00854 P. R. Cha agchuxua@sohu.com bsract 67 Frmat wrot: It s mpossbl to sparat a cub to two cubs or a bquadrat to two bquadrats or gral
More informationFermat Last Theorem And Riemann Hypothesis(3) Automorphic Functions And Fermat s Last Theorem(3) (Fermat s Proof of FLT)
Fermat Last Theorem Ad Rema Hypothess(3) Automorphc Fuctos Ad Fermat s Last Theorem(3) (Fermat s Proof of FLT) Chu-Xua Jag P. O. Box 394, Beg 854, P. R. Cha agchuxua@sohu.com Abstract I 637 Fermat wrote:
More informationCOMPLEX NUMBERS AND ELEMENTARY FUNCTIONS OF COMPLEX VARIABLES
COMPLEX NUMBERS AND ELEMENTARY FUNCTIONS OF COMPLEX VARIABLES DEFINITION OF A COMPLEX NUMBER: A umbr of th form, whr = (, ad & ar ral umbrs s calld a compl umbr Th ral umbr, s calld ral part of whl s calld
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 informationIn 1991 Fermat s Last Theorem Has Been Proved(II)
In 99 Fermat s Last Theorem Has Been Proved(II) Chun-Xuan Jang P. O. Box 9, Beng 0085, P. R. Chna cxuan00@sna.com Abstract In 67 Fermat wrote: It s mpossble to separate a cube nto two cubes, or a bquadrate
More informationThe Simplest Proof of Fermat Last Theorem(1)
The Smplest roof of Fermat Last Theorem( Chun-Xuan Jang. O. Bo 39, Beng 0085,. R. Chna 3angchunuan@gmal.com Abstract In 637 Fermat wrote: It s mpossble to separate a cube nto two cubes, or a bquadrate
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 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 information3.4 Properties of the Stress Tensor
cto.4.4 Proprts of th trss sor.4. trss rasformato Lt th compots of th Cauchy strss tsor a coordat systm wth bas vctors b. h compots a scod coordat systm wth bas vctors j,, ar gv by th tsor trasformato
More informationAlmost all Cayley Graphs Are Hamiltonian
Acta Mathmatca Sca, Nw Srs 199, Vol1, No, pp 151 155 Almost all Cayly Graphs Ar Hamltoa Mg Jxag & Huag Qogxag Abstract It has b cocturd that thr s a hamltoa cycl vry ft coctd Cayly graph I spt of th dffculty
More informationChiang Mai J. Sci. 2014; 41(2) 457 ( 2) ( ) ( ) forms a simply periodic Proof. Let q be a positive integer. Since
56 Chag Ma J Sc 0; () Chag Ma J Sc 0; () : 56-6 http://pgscccmuacth/joural/ Cotrbutd Papr Th Padova Sucs Ft Groups Sat Taș* ad Erdal Karaduma Dpartmt of Mathmatcs, Faculty of Scc, Atatürk Uvrsty, 50 Erzurum,
More informationComplex Numbers. Prepared by: Prof. Sunil Department of Mathematics NIT Hamirpur (HP)
th Topc Compl Nmbrs Hyprbolc fctos ad Ivrs hyprbolc fctos, Rlato btw hyprbolc ad crclar fctos, Formla of hyprbolc fctos, Ivrs hyprbolc fctos Prpard by: Prof Sl Dpartmt of Mathmatcs NIT Hamrpr (HP) Hyprbolc
More informationReliability of time dependent stress-strength system for various distributions
IOS Joural of Mathmatcs (IOS-JM ISSN: 78-578. Volum 3, Issu 6 (Sp-Oct., PP -7 www.osrjourals.org lablty of tm dpdt strss-strgth systm for varous dstrbutos N.Swath, T.S.Uma Mahswar,, Dpartmt of Mathmatcs,
More information07 - SEQUENCES AND SERIES Page 1 ( Answers at he end of all questions ) b, z = n
07 - SEQUENCES AND SERIES Pag ( Aswrs at h d of all qustios ) ( ) If = a, y = b, z = c, whr a, b, c ar i A.P. ad = 0 = 0 = 0 l a l
More informationUnbalanced Panel Data Models
Ubalacd Pal Data odls Chaptr 9 from Baltag: Ecoomtrc Aalyss of Pal Data 5 by Adrás alascs 4448 troducto balacd or complt pals: a pal data st whr data/obsrvatos ar avalabl for all crosssctoal uts th tr
More informationIntroduction to logistic regression
Itroducto to logstc rgrsso Gv: datast D { 2 2... } whr s a k-dmsoal vctor of ral-valud faturs or attrbuts ad s a bar class labl or targt. hus w ca sa that R k ad {0 }. For ampl f k 4 a datast of 3 data
More informationDifferent types of Domination in Intuitionistic Fuzzy Graph
Aals of Pur ad Appld Mathmatcs Vol, No, 07, 87-0 ISSN: 79-087X P, 79-0888ol Publshd o July 07 wwwrsarchmathscorg DOI: http://dxdoorg/057/apama Aals of Dffrt typs of Domato Itutostc Fuzzy Graph MGaruambga,
More informationRepeated Trials: We perform both experiments. Our space now is: Hence: We now can define a Cartesian Product Space.
Rpatd Trals: As w hav lood at t, th thory of probablty dals wth outcoms of sgl xprmts. I th applcatos o s usually trstd two or mor xprmts or rpatd prformac or th sam xprmt. I ordr to aalyz such problms
More informationIntegral points on hyperbolas over Z: A special case
Itgral pots o hprbolas ovr Z: A spcal cas `Pag of 7 Kostat Zlator Dpartmt of Mathmatcs ad Computr Scc Rhod Islad Collg 600 Mout Plasat Avu Provdc, R.I. 0908-99, U.S.A. -mal addrss: ) Kzlator@rc.du ) Kostat_zlator@ahoo.com
More informationERDOS-SMARANDACHE NUMBERS. Sabin Tabirca* Tatiana Tabirca**
ERDO-MARANDACHE NUMBER b Tbrc* Tt Tbrc** *Trslv Uvrsty of Brsov, Computr cc Dprtmt **Uvrsty of Mchstr, Computr cc Dprtmt Th strtg pot of ths rtcl s rprstd by rct work of Fch []. Bsd o two symptotc rsults
More informationLECTURE 6 TRANSFORMATION OF RANDOM VARIABLES
LECTURE 6 TRANSFORMATION OF RANDOM VARIABLES TRANSFORMATION OF FUNCTION OF A RANDOM VARIABLE UNIVARIATE TRANSFORMATIONS TRANSFORMATION OF RANDOM VARIABLES If s a rv wth cdf F th Y=g s also a rv. If w wrt
More informationCounting the compositions of a positive integer n using Generating Functions Start with, 1. x = 3 ), the number of compositions of 4.
Coutg th compostos of a postv tgr usg Gratg Fuctos Start wth,... - Whr, for ampl, th co-ff of s, for o summad composto of aml,. To obta umbr of compostos of, w d th co-ff of (...) ( ) ( ) Hr for stac w
More informationFurther Results on Pair Sum Graphs
Applid Mathmatis, 0,, 67-75 http://dx.doi.org/0.46/am.0.04 Publishd Oli Marh 0 (http://www.sirp.org/joural/am) Furthr Rsults o Pair Sum Graphs Raja Poraj, Jyaraj Vijaya Xavir Parthipa, Rukhmoi Kala Dpartmt
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 informationWorksheet: Taylor Series, Lagrange Error Bound ilearnmath.net
Taylor s Thorm & Lagrag Error Bouds Actual Error This is th ral amout o rror, ot th rror boud (worst cas scario). It is th dirc btw th actual () ad th polyomial. Stps:. Plug -valu ito () to gt a valu.
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 informationOn the approximation of the constant of Napier
Stud. Uiv. Babş-Bolyai Math. 560, No., 609 64 O th approximatio of th costat of Napir Adri Vrscu Abstract. Startig from som oldr idas of [] ad [6], w show w facts cocrig th approximatio of th costat of
More informationCorrelation in tree The (ferromagnetic) Ising model
5/3/00 :\ jh\slf\nots.oc\7 Chaptr 7 Blf propagato corrlato tr Corrlato tr Th (frromagtc) Isg mol Th Isg mol s a graphcal mol or par ws raom Markov fl cosstg of a urct graph wth varabls assocat wth th vrtcs.
More informationNEW VERSION OF SZEGED INDEX AND ITS COMPUTATION FOR SOME NANOTUBES
Digst Joural of Naomatrials ad Biostructurs Vol 4, No, March 009, p 67-76 NEW VERSION OF SZEGED INDEX AND ITS COMPUTATION FOR SOME NANOTUBES A IRANMANESH a*, O KHORMALI b, I NAJAFI KHALILSARAEE c, B SOLEIMANI
More informationThe probability of Riemann's hypothesis being true is. equal to 1. Yuyang Zhu 1
Th robablty of Ra's hyothss bg tru s ual to Yuyag Zhu Abstract Lt P b th st of all r ubrs P b th -th ( ) lt of P ascdg ordr of sz b ostv tgrs ad s a rutato of wth Th followg rsults ar gv ths ar: () Th
More informationMath Tricks. Basic Probability. x k. (Combination - number of ways to group r of n objects, order not important) (a is constant, 0 < r < 1)
Math Trcks r! Combato - umbr o was to group r o objcts, ordr ot mportat r! r! ar 0 a r a s costat, 0 < r < k k! k 0 EX E[XX-] + EX Basc Probablt 0 or d Pr[X > ] - Pr[X ] Pr[ X ] Pr[X ] - Pr[X ] Proprts
More informationMATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY. e S(A)/ da, h N
MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY 9 4. Matrx tgrals Lt h N b th spac of Hrmta matrcs of sz N. Th r product o h N s gv by (A, B) = Tr(AB). I ths scto w wll cosdr tgrals of th form Z
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 informationThe real E-k diagram of Si is more complicated (indirect semiconductor). The bottom of E C and top of E V appear for different values of k.
Modr Smcoductor Dvcs for Itgratd rcuts haptr. lctros ad Hols Smcoductors or a bad ctrd at k=0, th -k rlatoshp ar th mmum s usually parabolc: m = k * m* d / dk d / dk gatv gatv ffctv mass Wdr small d /
More informationz 1+ 3 z = Π n =1 z f() z = n e - z = ( 1-z) e z e n z z 1- n = ( 1-z/2) 1+ 2n z e 2n e n -1 ( 1-z )/2 e 2n-1 1-2n -1 1 () z
Sris Expasio of Rciprocal of Gamma Fuctio. Fuctios with Itgrs as Roots Fuctio f with gativ itgrs as roots ca b dscribd as follows. f() Howvr, this ifiit product divrgs. That is, such a fuctio caot xist
More informationSECTION where P (cos θ, sin θ) and Q(cos θ, sin θ) are polynomials in cos θ and sin θ, provided Q is never equal to zero.
SETION 6. 57 6. Evaluation of Dfinit Intgrals Exampl 6.6 W hav usd dfinit intgrals to valuat contour intgrals. It may com as a surpris to larn that contour intgrals and rsidus can b usd to valuat crtain
More informationHadamard Exponential Hankel Matrix, Its Eigenvalues and Some Norms
Math Sci Ltt Vol No 8-87 (0) adamard Exotial al Matrix, Its Eigvalus ad Som Norms İ ad M bula Mathmatical Scics Lttrs Itratioal Joural @ 0 NSP Natural Scics Publishig Cor Dartmt of Mathmatics, aculty of
More information1985 AP Calculus BC: Section I
985 AP Calculus BC: Sctio I 9 Miuts No Calculator Nots: () I this amiatio, l dots th atural logarithm of (that is, logarithm to th bas ). () Ulss othrwis spcifid, th domai of a fuctio f is assumd to b
More informationA COMPARISON OF SEVERAL TESTS FOR EQUALITY OF COEFFICIENTS IN QUADRATIC REGRESSION MODELS UNDER HETEROSCEDASTICITY
Colloquum Bomtrcum 44 04 09 7 COMPISON OF SEVEL ESS FO EQULIY OF COEFFICIENS IN QUDIC EGESSION MODELS UNDE HEEOSCEDSICIY Małgorzata Szczpa Dorota Domagała Dpartmt of ppld Mathmatcs ad Computr Scc Uvrsty
More informationNumerical Method: Finite difference scheme
Numrcal Mthod: Ft dffrc schm Taylor s srs f(x 3 f(x f '(x f ''(x f '''(x...(1! 3! f(x 3 f(x f '(x f ''(x f '''(x...(! 3! whr > 0 from (1, f(x f(x f '(x R Droppg R, f(x f(x f '(x Forward dffrcg O ( x from
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 informationBinary Choice. Multiple Choice. LPM logit logistic regresion probit. Multinomial Logit
(c Pogsa Porchawssul, Faculty of Ecoomcs, Chulalogor Uvrsty (c Pogsa Porchawssul, Faculty of Ecoomcs, Chulalogor Uvrsty 3 Bary Choc LPM logt logstc rgrso probt Multpl Choc Multomal Logt (c Pogsa Porchawssul,
More informationLecture 1: Empirical economic relations
Ecoomcs 53 Lctur : Emprcal coomc rlatos What s coomtrcs? Ecoomtrcs s masurmt of coomc rlatos. W d to kow What s a coomc rlato? How do w masur such a rlato? Dfto: A coomc rlato s a rlato btw coomc varabls.
More informationIranian Journal of Mathematical Chemistry, Vol. 2, No. 2, December 2011, pp (Received September 10, 2011) ABSTRACT
Iraa Joral of Mathatcal Chstry Vol No Dcbr 0 09 7 IJMC Two Tys of Gotrc Arthtc dx of V hylc Naotb S MORADI S BABARAHIM AND M GHORBANI Dartt of Mathatcs Faclty of Scc Arak Ursty Arak 856-8-89 I R Ira Dartt
More informationPURE MATHEMATICS A-LEVEL PAPER 1
-AL P MATH PAPER HONG KONG EXAMINATIONS AUTHORITY HONG KONG ADVANCED LEVEL EXAMINATION PURE MATHEMATICS A-LEVEL PAPER 8 am am ( hours) This papr must b aswrd i Eglish This papr cosists of Sctio A ad Sctio
More informationSTIRLING'S 1 FORMULA AND ITS APPLICATION
MAT-KOL (Baja Luka) XXIV ()(08) 57-64 http://wwwimviblorg/dmbl/dmblhtm DOI: 075/МК80057A ISSN 0354-6969 (o) ISSN 986-588 (o) STIRLING'S FORMULA AND ITS APPLICATION Šfkt Arslaagić Sarajvo B&H Abstract:
More informationThree-Dimensional Theory of Nonlinear-Elastic. Bodies Stability under Finite Deformations
Appld Mathmatcal Sccs ol. 9 5 o. 43 75-73 HKAR Ltd www.m-hkar.com http://dx.do.org/.988/ams.5.567 Thr-Dmsoal Thory of Nolar-Elastc Bods Stablty udr Ft Dformatos Yu.. Dmtrko Computatoal Mathmatcs ad Mathmatcal
More informationExercises for Square-Congruence Modulo n ver 11
Exercses for Square-Cogruece Modulo ver Let ad ab,.. Mark True or False. a. 3S 30 b. 3S 90 c. 3S 3 d. 3S 4 e. 4S f. 5S g. 0S 55 h. 8S 57. 9S 58 j. S 76 k. 6S 304 l. 47S 5347. Fd the equvalece classes duced
More informationChapter Five. More Dimensions. is simply the set of all ordered n-tuples of real numbers x = ( x 1
Chatr Fiv Mor Dimsios 51 Th Sac R W ar ow rard to mov o to sacs of dimsio gratr tha thr Ths sacs ar a straightforward gralizatio of our Euclida sac of thr dimsios Lt b a ositiv itgr Th -dimsioal Euclida
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 informationChapter 4 NUMERICAL METHODS FOR SOLVING BOUNDARY-VALUE PROBLEMS
Chaptr 4 NUMERICL METHODS FOR SOLVING BOUNDRY-VLUE PROBLEMS 00 4. Varatoal formulato two-msoal magtostatcs Lt th followg magtostatc bouar-valu problm b cosr ( ) J (4..) 0 alog ΓD (4..) 0 alog ΓN (4..)
More informationChapter 6. pn-junction diode: I-V characteristics
Chatr 6. -jucto dod: -V charactrstcs Tocs: stady stat rsos of th jucto dod udr ald d.c. voltag. ucto udr bas qualtatv dscusso dal dod quato Dvatos from th dal dod Charg-cotrol aroach Prof. Yo-S M Elctroc
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 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 informationTriple Play: From De Morgan to Stirling To Euler to Maclaurin to Stirling
Tripl Play: From D Morga to Stirlig To Eulr to Maclauri to Stirlig Augustus D Morga (186-1871) was a sigificat Victoria Mathmaticia who mad cotributios to Mathmatics History, Mathmatical Rcratios, Mathmatical
More informationEntropy Equation for a Control Volume
Fudamtals of Thrmodyamcs Chaptr 7 Etropy Equato for a Cotrol Volum Prof. Syoug Jog Thrmodyamcs I MEE2022-02 Thrmal Egrg Lab. 2 Q ds Srr T Q S2 S1 1 Q S S2 S1 Srr T t t T t S S s m 1 2 t S S s m tt S S
More informationCramér-Rao Inequality: Let f(x; θ) be a probability density function with continuous parameter
WHEN THE CRAMÉR-RAO INEQUALITY PROVIDES NO INFORMATION STEVEN J. MILLER Abstract. W invstigat a on-paramtr family of probability dnsitis (rlatd to th Parto distribution, which dscribs many natural phnomna)
More informationOn Estimation of Unknown Parameters of Exponential- Logarithmic Distribution by Censored Data
saqartvlos mcrbata rovul akadms moamb, t 9, #2, 2015 BULLETIN OF THE GEORGIAN NATIONAL ACADEMY OF SCIENCES, vol 9, o 2, 2015 Mathmatcs O Estmato of Ukow Paramtrs of Epotal- Logarthmc Dstrbuto by Csord
More informationPerfect Constant-Weight Codes
56 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 9, SEPTEMBER 004 Prfct Costat-Wght Cods Tuv Etzo, Fllo, IEEE, ad Mosh Schartz, Mmbr, IEEE Abstract I hs porg or from 973, Dlsart cocturd that thr
More informationGraphs of q-exponentials and q-trigonometric functions
Grahs of -otals ad -trgoomtrc fuctos Amla Carola Saravga To ct ths vrso: Amla Carola Saravga. Grahs of -otals ad -trgoomtrc fuctos. 26. HAL Id: hal-377262 htts://hal.archvs-ouvrts.fr/hal-377262
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 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 informationOn the irreducibility of some polynomials in two variables
ACTA ARITHMETICA LXXXII.3 (1997) On th irrducibility of som polynomials in two variabls by B. Brindza and Á. Pintér (Dbrcn) To th mmory of Paul Erdős Lt f(x) and g(y ) b polynomials with intgral cofficints
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 informationDSP-First, 2/e. LECTURE # CH2-3 Complex Exponentials & Complex Numbers TLH MODIFIED. Aug , JH McClellan & RW Schafer
DSP-First, / TLH MODIFIED LECTURE # CH-3 Complx Exponntials & Complx Numbrs Aug 016 1 READING ASSIGNMENTS This Lctur: Chaptr, Scts. -3 to -5 Appndix A: Complx Numbrs Complx Exponntials Aug 016 LECTURE
More informationSolution to 1223 The Evil Warden.
Solutio to 1 Th Evil Ward. This is o of thos vry rar PoWs (I caot thik of aothr cas) that o o solvd. About 10 of you submittd th basic approach, which givs a probability of 47%. I was shockd wh I foud
More informationStatistics 3858 : Likelihood Ratio for Exponential Distribution
Statistics 3858 : Liklihood Ratio for Expotial Distributio I ths two xampl th rjctio rjctio rgio is of th form {x : 2 log (Λ(x)) > c} for a appropriat costat c. For a siz α tst, usig Thorm 9.5A w obtai
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 informationThomas J. Osler. 1. INTRODUCTION. This paper gives another proof for the remarkable simple
5/24/5 A PROOF OF THE CONTINUED FRACTION EXPANSION OF / Thomas J Oslr INTRODUCTION This ar givs aothr roof for th rmarkabl siml cotiud fractio = 3 5 / Hr is ay ositiv umbr W us th otatio x= [ a; a, a2,
More information1973 AP Calculus BC: Section I
97 AP Calculus BC: Scio I 9 Mius No Calculaor No: I his amiaio, l dos h aural logarihm of (ha is, logarihm o h bas ).. If f ( ) =, h f ( ) = ( ). ( ) + d = 7 6. If f( ) = +, h h s of valus for which f
More informationA Review of Complex Arithmetic
/0/005 Rviw of omplx Arithmti.do /9 A Rviw of omplx Arithmti A omplx valu has both a ral ad imagiary ompot: { } ad Im{ } a R b so that w a xprss this omplx valu as: whr. a + b Just as a ral valu a b xprssd
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 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 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 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 informationChapter Discrete Fourier Transform
haptr.4 Dscrt Fourr Trasform Itroducto Rcad th xpota form of Fourr srs s Equatos 8 ad from haptr., wt f t 8, h.. T w t f t dt T Wh th abov tgra ca b usd to comput, h.., t s mor prfrab to hav a dscrtzd
More informationNew families of p-ary sequences with low correlation and large linear span
THE JOURNAL OF CHINA UNIVERSITIES OF POSTS AND TELECOMMUNICATIONS Volu 4 Issu 4 Dcbr 7 TONG X WEN Qao-ya Nw fals of -ary sucs wth low corrlato ad larg lar sa CLC ubr TN98 Docut A Artcl ID 5-8885 (7 4-53-4
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 informationNuclear Chemistry -- ANSWERS
Hoor Chstry Mr. Motro 5-6 Probl St Nuclar Chstry -- ANSWERS Clarly wrt aswrs o sparat shts. Show all work ad uts.. Wrt all th uclar quatos or th radoactv dcay srs o Urau-38 all th way to Lad-6. Th dcay
More informationNote: Torque is prop. to current Stationary voltage is prop. to speed
DC Mach Cotrol Mathmatcal modl. Armatr ad orq f m m a m m r a a a a a dt d ψ ψ ψ ω Not: orq prop. to crrt Statoary voltag prop. to pd Mathmatcal modl. Fld magtato f f f f d f dt a f ψ m m f f m fλ h torq
More informationSecond Handout: The Measurement of Income Inequality: Basic Concepts
Scod Hadout: Th Masurmt of Icom Iqualty: Basc Cocpts O th ormatv approach to qualty masurmt ad th cocpt of "qually dstrbutd quvalt lvl of com" Suppos that that thr ar oly two dvduals socty, Rachl ad Mart
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 informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!
More informationCarbonyl Groups. University of Chemical Technology, Beijing , PR China;
Electroc Supplemetary Materal (ESI) for Physcal Chemstry Chemcal Physcs Ths joural s The Ower Socetes 0 Supportg Iformato A Theoretcal Study of Structure-Solublty Correlatos of Carbo Doxde Polymers Cotag
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 informationChapter 14 Logistic Regression Models
Chapter 4 Logstc Regresso Models I the lear regresso model X β + ε, there are two types of varables explaatory varables X, X,, X k ad study varable y These varables ca be measured o a cotuous scale as
More informationChapter Taylor Theorem Revisited
Captr 0.07 Taylor Torm Rvisitd Atr radig tis captr, you sould b abl to. udrstad t basics o Taylor s torm,. writ trascdtal ad trigoomtric uctios as Taylor s polyomial,. us Taylor s torm to id t valus o
More informationMath 102. Rumbos Spring Solutions to Assignment #8. Solution: The matrix, A, corresponding to the system in (1) is
Math 12. Rumbos Spring 218 1 Solutions to Assignmnt #8 1. Construct a fundamntal matrix for th systm { ẋ 2y ẏ x + y. (1 Solution: Th matrix, A, corrsponding to th systm in (1 is 2 A. (2 1 1 Th charactristic
More informationChapter 2 Infinite Series Page 1 of 11. Chapter 2 : Infinite Series
Chatr Ifiit Sris Pag of Sctio F Itgral Tst Chatr : Ifiit Sris By th d of this sctio you will b abl to valuat imror itgrals tst a sris for covrgc by alyig th itgral tst aly th itgral tst to rov th -sris
More informationIdeal multigrades with trigonometric coefficients
Ideal multgrades wth trgoometrc coeffcets Zarathustra Brady December 13, 010 1 The problem A (, k) multgrade s defed as a par of dstct sets of tegers such that (a 1,..., a ; b 1,..., b ) a j = =1 for all
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!
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 informationInternational Journal of Mathematical Archive-6(5), 2015, Available online through ISSN
Itratoal Joural of Mathmatal Arhv-6), 0, 07- Avalabl ol through wwwjmafo ISSN 9 06 ON THE LINE-CUT TRANSFORMATION RAPHS B BASAVANAOUD*, VEENA R DESAI Dartmt of Mathmats, Karatak Uvrsty, Dharwad - 80 003,
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 informationSuzan Mahmoud Mohammed Faculty of science, Helwan University
Europa Joural of Statstcs ad Probablty Vol.3, No., pp.4-37, Ju 015 Publshd by Europa Ctr for Rsarch Trag ad Dvlopmt UK (www.ajourals.org ESTIMATION OF PARAMETERS OF THE MARSHALL-OLKIN WEIBULL DISTRIBUTION
More informationA Measure of Inaccuracy between Two Fuzzy Sets
LGRN DEMY OF SENES YERNETS ND NFORMTON TEHNOLOGES Volum No 2 Sofa 20 Masur of accuracy btw Two Fuzzy Sts Rajkumar Vrma hu Dv Sharma Dpartmt of Mathmatcs Jayp sttut of formato Tchoy (Dmd vrsty) Noda (.P.)
More informationThe Interplay between l-max, l-min, p-max and p-min Stable Distributions
DOI: 0.545/mjis.05.4006 Th Itrplay btw lma lmi pma ad pmi Stabl Distributios S Ravi ad TS Mavitha Dpartmt of Studis i Statistics Uivrsity of Mysor Maasagagotri Mysuru 570006 Idia. Email:ravi@statistics.uimysor.ac.i
More informationNP!= P. By Liu Ran. Table of Contents. The P versus NP problem is a major unsolved problem in computer
NP!= P By Lu Ra Table of Cotets. Itroduce 2. Prelmary theorem 3. Proof 4. Expla 5. Cocluso. Itroduce The P versus NP problem s a major usolved problem computer scece. Iformally, t asks whether a computer
More informationIntroduction to logistic regression
Itroducto to logstc rgrsso Gv: datast D {... } whr s a k-dmsoal vctor of ral-valud faturs or attrbuts ad s a bar class labl or targt. hus w ca sa that R k ad {0 }. For ampl f k 4 a datast of 3 data pots
More informationWashington State University
he 3 Ktics ad Ractor Dsig Sprg, 00 Washgto Stat Uivrsity Dpartmt of hmical Egrg Richard L. Zollars Exam # You will hav o hour (60 muts) to complt this xam which cosists of four (4) problms. You may us
More information