The Simplest Proof of Fermat Last Theorem(1)
|
|
|
- Julia Johns
- 5 years ago
- Views:
Transcription
1 The Smplest roof of Fermat Last Theorem( Chun-Xuan Jang. O. Bo 39, Beng 0085,. R. Chna Abstract In 637 Fermat wrote: It s mpossble to separate a cube nto two cubes, or a bquadrate nto two bquadrates, or n general any power hgher than the second nto powers of lke degree: I have dscovered a truly marvelous proof, whch ths margn s too small to contan. n n n Ths means: y z ( n has no nteger solutons, all dfferent from 0(.e., t has only the trval soluton, where one of the ntegers s equal to 0. It has been called Fermat s last theorem (FLT. It suffces to prove FLT for eponent and every prme eponent. Fermat proved FLT for eponent. Euler proved FLT for eponent 3. In ths paper usng automorphc functons we prove FLT for eponents and, where s an odd prme. We redscover the Fermat proof. The proof of FLT must be drect. But ndrect proof of FLT s dsbelevng. Theorem:The smplest proof of Fermat last theorem.we have Fermat equaton y z, where s odd prme. We prove that f y and z are nteger numbers then, and are rratonal numbers. In 97 Jang found out Euler formula of the cyclotomc real numbers n the cyclotomc felds ep where J denotes a m th root of unty, S m m tj SJ, ( m J, m=,,3,, t are the real numbers. s called the automorphc functons(comple hyperbolc functons of order m wth m varables [,5,7]. ( ( cos cos m m A H B S e e e m
2 ( m ( A D ( e e cos m m ( where,...,m; m m A t, A t(, H t (, t ( m m B t cos, t sn, m m D t ( cos, t( sn, m m m m m m, m A A H ( B D 0. (3 From ( we have ts nverse transformaton[5,7] m m A A, ( e S e S m m H H e cos S (, e sn S (, B e cos S S cos, e sn S sn m m m B m, D e cos S S ( cos, e sn S ( sn m m m D m. ( (3 and ( have the same form. From (3 we have m ep A A H ( B D From ( we have S S S m m S S S3 ep A A H ( B D S S S m m (5
3 where S ( S [7] t S ( S ( S m S ( S ( S m S ( S ( S m m m m (6 From (5 and (6 we have crculant determnant S S S m S S S m 3 ep A A H ( B D S S S m m (7 Assume S 0, S 0, S 0, where 3,..., m. S 0 are (m ndetermnate equatons wth (m varables. From ( we have A A H e SS, e SS, e S S B D e S S SScos, e S S SScos (8 m m Eample []. Let m. From (3 we have A ( tt ( t t0 ( t3t9 ( t t8 ( t5 t7 t 6, A ( t t ( t t ( t t ( t t ( t t t 6, H ( t t ( t t t, B ( ttcos ( t t0cos ( t3 t9cos ( t t8cos ( t5 t7cos t6, B ( tt cos ( t t0 cos ( t3t9 cos ( t t8 cos ( t5 t7 cos t6, 3 5 D ( tt cos ( t t0 cos ( t3t9 cos ( t t8cos ( t5 t7 cos t6, D ( ttcos ( t t0cos ( t3t9cos ( t t8cos ( t5 t7cos t6, A A ( H BB DD 0, A B 3( t3 t6 t9. (9 From (8 and (9 we have 3 3 ep[ A A ( H B B D D ] S S ( S ( S. (0 From (9 we have 3
4 3 ep( A B [ep( t t t ]. ( From (8 we have ep( A B ( S S ( S S S S S S. ( 3 From ( and ( we have Fermat s equaton ep( A B S S [ep( t t t ] 3. ( Fermat proved that (0 has no ratonal solutons for eponent [8]. Therefore we prove we prove that (3 has no ratonal solutons for eponent 3. [] Theorem. Let m, where s an odd prme, ( / s an even number. From (3 and (8 we have From (3 we have From (8 we have ep[ A A H ( B D ] S S ( S ( S. ( ep[ A ( B D ] [ep( t t t3 ]. (5 S ep[ A ( B D ] S From (5 and (6 we have Fermat s equaton 3. (6 3 ep[ A ( B D ] S S [ep( t t t ]. (7 Fermat proved that ( has no ratonal solutons for eponent [8]. Therefor we prove that (7 has no ratonal solutons for prme eponent. Theorem.The smplest proof of Fermat last theorem. We have the Fermat equaton y z (8 where s the odd prme. We rewrte (8 ( ( y ( z. (9 Fermat proved that (9 has no nteger solutons for eponent. We assume that y and z are nteger numbers, y and z also are nteger numbers. We have and are rratonal
5 numbers. Therefore we prove (8 has no nteger solutons We rewrte (8 ( ( y ( z (0 We assume that y and z are nteger numbers, y and z also are nteger numbers. From theorem we prove s rratonal number, also s rratonal number. Therefore we prove (0 has no nteger solutons for prme eponent. Concluson.In(8 we prove that f y and then, and are rratonal numbers.. z are nteger numbers References [] Jang, C-X, Fermat last theorem had been proved, otental Scence (n Chnese,.7-0 (99, reprnts (n Englsh December (99. [] Jang, C-X, Fermat last theorem had been proved by Fermat more than 300 years ago, otental Scence (n Chnese, 6.8-0(99. [3] Jang, C-X, On the factorzaton theorem of crculant determnant, Algebras, Groups and Geometres, (99, MR. 96a: 03, [] Jang, C-X, Fermat last theorem was proved n 99, reprnts (993. In: Fundamental open problems n scence at the end of the mllennum, T.Gll, K. Lu and E. Trell (eds. Hadronc ress, 999, [5] Jang, C-X, On the Fermat-Santll theorem, Algebras, Groups and Geometres, (998 [6] Jang, C-X, Comple hyperbolc functons and Fermat s last theorem, Hadronc Journal Supplement, (000. [7] Jang, C-X, Foundatons of Santll Isonumber Theory wth applcatons to new cryptograms, Fermat s theorem and Goldbach s Conecture. Inter. Acad. ress. 00. MR00c:00, [8] Rbenbom,. Fermat last theorem for amateur,sprnger-verlag(999. 5
In 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
( N) Chun-Xuan Jiang. P. O. Box 3924, Beijing , P. R. China
ang s functon n ( ) n prme dstrbuton Chun-Xuan ang P O Box 94, Bejng 00854, P R Chna jcxuan@snacom Abstract: We defne that prme equatons f( P,, Pn ),, f ( P, Pn ) (5)are polynomals (wth nteger coeffcents)
On the singular series in the Jiang prime k-tuple theorem
On the sngular seres n the Jang prme -tuple theorem Chun-Xuan Jang. O. Box 94, Bejng 10084,. R. Chna jcxuan@sna.com Abstract Usng Jang functon we prove Jang prme -tuple theorem.we fnd true sngular seres.
The Hardy-Littlewood prime k-tuple conjecture is false
The Hardy-Lttlewood prme k-tuple conjecture s false Chun-Xuan Jang. O. Box 9, Bejng 008,. R. Chna Jangchunxuan@vp.sohu.com Abstract Usng Jang functon we prove Jang prme k -tuple theorem. We prove that
On the singular series in the Jiang prime k-tuple theorem
On the sngular seres n the Jang prme k-tuple theorem Chun-Xuan Jang. O. Box 9, Bejng 1008,. R. Chna jcxuan@sna.com Abstract Usng Jang functon we prove Jang prme k -tuple theorem.we fnd true sngular seres.
GENERALIZED ARITHMETIC PROGRESSIONS
GENERALIZED ARITHMETIC ROGRESSIONS AND k Chun-Xuan an Bejn 00854,. O. Box 94,. R. Chna Lukx@publc.bta.net.cn Abstract We defne the eneralzed arthetc proressons and k, where,, k,,. If the arthetc functon
Fermat s Marvelous Proofs for Fermat s Last Theorem. Abstract
Fermat s Marvelous Proofs for Fermat s Last Theorem Chun-Xuan, Jiang Institute for Basic Research Palm Harbor, FL468-577, USA and P. O. Box 94, Beijing 00854, China jiangchunxuan@sohu.com Abstract Using
The Hardy-Littlewood prime k-tuple conjecture is false
The Hardy-Lttlewood prme -tuple conjecture s false Chun-Xuan Jang. O. Box 9, Bejng 008,. R. Chna Jangchunxuan@vp.sohu.com Abstract Usng Jang functon we prove Jang prme -tuple theorem. We prove that the
Fermat 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:
In 1991 Fermat s Last Theorem Has Been Proved
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
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
Euclid-Euler-Jiang Prime Theorem
Euclid-Euler-iang Prime Theorem Chun-Xuan iang POBox 394 Beiing 00854 P R China iangchunxuan@sohucom Abstract Santilli s prime chains: P + ap b L ( a b ab If λn a P λ LP n P LPn b we have ( as There exist
Bernoulli Numbers and Polynomials
Bernoull Numbers and Polynomals T. Muthukumar tmk@tk.ac.n 17 Jun 2014 The sum of frst n natural numbers 1, 2, 3,..., n s n n(n + 1 S 1 (n := m = = n2 2 2 + n 2. Ths formula can be derved by notng that
18.781: Solution to Practice Questions for Final Exam
18.781: Soluton to Practce Questons for Fnal Exam 1. Fnd three solutons n postve ntegers of x 6y = 1 by frst calculatng the contnued fracton expanson of 6. Soluton: We have 1 6=[, ] 6 6+ =[, ] 1 =[,, ]=[,,
Riemann s Hypothesis and Conjecture of Birch and Swinnerton-Dyer are False
Riemann s Hypothesis and Conjecture of Birch and Swinnerton-yer are False Chun-Xuan Jiang. O. Box 3924, Beijing 854 China jcxuan@sina.com Abstract All eyes are on the Riemann s hypothesis, zeta and L-functions,
Example: (13320, 22140) =? Solution #1: The divisors of are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 41,
The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no confuson
Unit 5: Quadratic Equations & Functions
Date Perod Unt 5: Quadratc Equatons & Functons DAY TOPIC 1 Modelng Data wth Quadratc Functons Factorng Quadratc Epressons 3 Solvng Quadratc Equatons 4 Comple Numbers Smplfcaton, Addton/Subtracton & Multplcaton
12. The Hamilton-Jacobi Equation Michael Fowler
1. The Hamlton-Jacob Equaton Mchael Fowler Back to Confguraton Space We ve establshed that the acton, regarded as a functon of ts coordnate endponts and tme, satsfes ( ) ( ) S q, t / t+ H qpt,, = 0, and
New Exact Traveling Wave Solutions for Two Nonlinear Evolution Equations
Internatonal Conference on Computer Technology and Scence (ICCTS ) IPCSIT vol. 47 () () IACSIT Press, Sngapore DOI:.7763/IPCSIT..V47.66 New Exact Travelng Wave Solutons for Two Nonlnear Evoluton Equatons
THE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens
THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of
Section 8.3 Polar Form of Complex Numbers
80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the
International Conference on Advanced Computer Science and Electronics Information (ICACSEI 2013) equation. E. M. E. Zayed and S. A.
Internatonal Conference on Advanced Computer Scence and Electroncs Informaton (ICACSEI ) The two varable (G'/G/G) -expanson method for fndng exact travelng wave solutons of the (+) dmensonal nonlnear potental
Section 3.6 Complex Zeros
04 Chapter Secton 6 Comple Zeros When fndng the zeros of polynomals, at some pont you're faced wth the problem Whle there are clearly no real numbers that are solutons to ths equaton, leavng thngs there
Pre-Calculus Summer Assignment
Pre-Calculus Summer Assgnment Dear Future Pre-Calculus Student, Congratulatons on our successful completon of Algebra! Below ou wll fnd the summer assgnment questons. It s assumed that these concepts,
9. Complex Numbers. 1. Numbers revisited. 2. Imaginary number i: General form of complex numbers. 3. Manipulation of complex numbers
9. Comple Numbers. Numbers revsted. Imagnar number : General form of comple numbers 3. Manpulaton of comple numbers 4. The Argand dagram 5. The polar form for comple numbers 9.. Numbers revsted We saw
Neumann Asymptotic Eigenvalues of Sturm-liouville Problem with Three Turning Points
Australan Journal of Basc and Appled Scences 5(5): 89-96 ISSN 99-878 Neumann Asymptotc Egenalues of Sturm-loulle Problem wth Three Turnng Ponts E.A.Sazgar Department of Mathematcs Yerean state Unersty
Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
Numerical Solution of Ordinary Differential Equations
Numercal Methods (CENG 00) CHAPTER-VI Numercal Soluton of Ordnar Dfferental Equatons 6 Introducton Dfferental equatons are equatons composed of an unknown functon and ts dervatves The followng are examples
Multiple-soliton Solutions for Nonlinear Partial Differential Equations
Journal of Mathematcs Research; Vol. 7 No. ; ISSN 9-979 E-ISSN 9-989 Publshed b Canadan Center of Scence and Educaton Multple-solton Solutons for Nonlnear Partal Dfferental Equatons Yanng Tang & Wean Za
Y. Guo. A. Liu, T. Liu, Q. Ma UDC
UDC 517. 9 OSCILLATION OF A CLASS OF NONLINEAR PARTIAL DIFFERENCE EQUATIONS WITH CONTINUOUS VARIABLES* ОСЦИЛЯЦIЯ КЛАСУ НЕЛIНIЙНИХ ЧАСТКОВО РIЗНИЦЕВИХ РIВНЯНЬ З НЕПЕРЕРВНИМИ ЗМIННИМИ Y. Guo Graduate School
Workshop: Approximating energies and wave functions Quantum aspects of physical chemistry
Workshop: Approxmatng energes and wave functons Quantum aspects of physcal chemstry http://quantum.bu.edu/pltl/6/6.pdf Last updated Thursday, November 7, 25 7:9:5-5: Copyrght 25 Dan Dll (dan@bu.edu) Department
SL 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
International Journal of Pure and Applied Mathematics Volume 17 No , EVALUATIONS OF THE IMPROPER INTEGRALS. sin 2m+1 (αx) x 2n+1
Internatonal Journal of Pure and Appled Mathematcs Volume 17 No. 3 4, 91-99 EVALUATIONS OF THE IMPROPER INTEGRALS sn m (α) AND n Qu-Mng Luo sn m+1 (α) Department of Mathematcs Jaozuo Unversty Jaozuo Cty,
MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS
MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS These are nformal notes whch cover some of the materal whch s not n the course book. The man purpose s to gve a number of nontrval examples
Polynomials. 1 What is a polynomial? John Stalker
Polynomals John Stalker What s a polynomal? If you thnk you already know what a polynomal s then skp ths secton. Just be aware that I consstently wrte thngs lke p = c z j =0 nstead of p(z) = c z. =0 You
Polynomial Regression Models
LINEAR REGRESSION ANALYSIS MODULE XII Lecture - 6 Polynomal Regresson Models Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur Test of sgnfcance To test the sgnfcance
Journal of Number Theory. On Euler numbers, polynomials and related p-adic integrals
Journal of Number Theory 19 009 166 179 Contents lsts avalable at ScenceDrect Journal of Number Theory www.elsever.com/locate/jnt On Euler numbers, polynomals and related p-adc ntegrals Mn-Soo Km Natonal
The Ramanujan-Nagell Theorem: Understanding the Proof By Spencer De Chenne
The Ramanujan-Nagell Theorem: Understandng the Proof By Spencer De Chenne 1 Introducton The Ramanujan-Nagell Theorem, frst proposed as a conjecture by Srnvasa Ramanujan n 1943 and later proven by Trygve
The internal structure of natural numbers and one method for the definition of large prime numbers
The nternal structure of natural numbers and one method for the defnton of large prme numbers Emmanul Manousos APM Insttute for the Advancement of Physcs and Mathematcs 3 Poulou str. 53 Athens Greece Abstract
Problem Solving in Math (Math 43900) Fall 2013
Problem Solvng n Math (Math 43900) Fall 2013 Week four (September 17) solutons Instructor: Davd Galvn 1. Let a and b be two nteger for whch a b s dvsble by 3. Prove that a 3 b 3 s dvsble by 9. Soluton:
A summation on Bernoulli numbers
Journal of Number Theory 111 (005 37 391 www.elsever.com/locate/jnt A summaton on Bernoull numbers Kwang-Wu Chen Department of Mathematcs and Computer Scence Educaton, Tape Muncpal Teachers College, No.
Salmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2
Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to
2-π STRUCTURES ASSOCIATED TO THE LAGRANGIAN MECHANICAL SYSTEMS UDC 531.3: (045)=111. Victor Blãnuţã, Manuela Gîrţu
FACTA UNIVERSITATIS Seres: Mechancs Automatc Control and Robotcs Vol. 6 N o 1 007 pp. 89-95 -π STRUCTURES ASSOCIATED TO THE LAGRANGIAN MECHANICAL SYSTEMS UDC 531.3:53.511(045)=111 Vctor Blãnuţã Manuela
ON THE ARGUMENT OF ABEL. William Rowan Hamilton
ON THE ARGUMENT OF ABEL By Wllam Rowan Hamlton (Transactons of the Royal Irsh Academy, 8 (839), pp. 7 259.) Edted by Davd R. Wlkns 2000 NOTE ON THE TEXT The text of ths edton s taken from the 8th volume
A P PL I CA TIONS OF FRACTIONAL EXTERIOR DI F F ER EN TIAL IN THR EE- DI M ENSIONAL S PAC E Ξ
Appled Mathematcs and Mechancs ( Englsh Edton, Vol 24, No 3, Mar 2003) Publshed by Shangha Unversty, Shangha, Chna Artcle ID : 0253-4827 (2003) 03-0256-05 A P PL I CA TIONS OF FRACTIONAL EXTERIOR DI F
UNIT 4 EXTENDING THE NUMBER SYSTEM Lesson 3: Operating with Complex Numbers Instruction
Prerequste Sklls Ths lesson requres the use of the followng sklls: understandng that multplyng the numerator and denomnator of a fracton by the same quantty produces an equvalent fracton multplyng complex
ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EQUATION
Advanced Mathematcal Models & Applcatons Vol.3, No.3, 2018, pp.215-222 ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EUATION
Computing π with Bouncing Balls
by Let the mass of two balls be M and m, where M = (6 n )m for n N. The larger ball s rolled towards the lghter ball, whch ear a wall and ntally at rest. Fnd the number of collsons between the two balls
MEM 255 Introduction to Control Systems Review: Basics of Linear Algebra
MEM 255 Introducton to Control Systems Revew: Bascs of Lnear Algebra Harry G. Kwatny Department of Mechancal Engneerng & Mechancs Drexel Unversty Outlne Vectors Matrces MATLAB Advanced Topcs Vectors A
Attacks on RSA The Rabin Cryptosystem Semantic Security of RSA Cryptology, Tuesday, February 27th, 2007 Nils Andersen. Complexity Theoretic Reduction
Attacks on RSA The Rabn Cryptosystem Semantc Securty of RSA Cryptology, Tuesday, February 27th, 2007 Nls Andersen Square Roots modulo n Complexty Theoretc Reducton Factorng Algorthms Pollard s p 1 Pollard
Implicit Integration Henyey Method
Implct Integraton Henyey Method In realstc stellar evoluton codes nstead of a drect ntegraton usng for example the Runge-Kutta method one employs an teratve mplct technque. Ths s because the structure
Exhaustive Search for the Binary Sequences of Length 2047 and 4095 with Ideal Autocorrelation
Exhaustve Search for the Bnary Sequences of Length 047 and 4095 wth Ideal Autocorrelaton 003. 5. 4. Seok-Yong Jn and Hong-Yeop Song. Yonse Unversty Contents Introducton Background theory Ideal autocorrelaton
For all questions, answer choice E) NOTA" means none of the above answers is correct.
0 MA Natonal Conventon For all questons, answer choce " means none of the above answers s correct.. In calculus, one learns of functon representatons that are nfnte seres called power 3 4 5 seres. For
Math 261 Exercise sheet 2
Math 261 Exercse sheet 2 http://staff.aub.edu.lb/~nm116/teachng/2017/math261/ndex.html Verson: September 25, 2017 Answers are due for Monday 25 September, 11AM. The use of calculators s allowed. Exercse
COMPLEX NUMBERS AND QUADRATIC EQUATIONS
COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not
NEW EXACT ANALYTICAL SOLUTIONS FOR THE GENERAL KDV EQUATION WITH VARIABLE COEFFICIENTS. Jiangsu, PR China
athematcal and Computatonal Applcatons Vol. 19 No. pp. 19-7 1 NEW EXACT ANALYTICAL SOLUTIONS FOR THE GENERAL KDV EQUATION WITH VARIABLE COEFFICIENTS Bao-Jan Hong 1 and Dan-Chen Lu 1* 1 Faculty of Scence
PART 8. Partial Differential Equations PDEs
he Islamc Unverst of Gaza Facult of Engneerng Cvl Engneerng Department Numercal Analss ECIV 3306 PAR 8 Partal Dfferental Equatons PDEs Chapter 9; Fnte Dfference: Ellptc Equatons Assocate Prof. Mazen Abualtaef
Quantum Mechanics I - Session 4
Quantum Mechancs I - Sesson 4 Aprl 3, 05 Contents Operators Change of Bass 4 3 Egenvectors and Egenvalues 5 3. Denton....................................... 5 3. Rotaton n D....................................
DISCRIMINANTS AND RAMIFIED PRIMES. 1. Introduction A prime number p is said to be ramified in a number field K if the prime ideal factorization
DISCRIMINANTS AND RAMIFIED PRIMES KEITH CONRAD 1. Introducton A prme number p s sad to be ramfed n a number feld K f the prme deal factorzaton (1.1) (p) = po K = p e 1 1 peg g has some e greater than 1.
The Gaussian classifier. Nuno Vasconcelos ECE Department, UCSD
he Gaussan classfer Nuno Vasconcelos ECE Department, UCSD Bayesan decson theory recall that we have state of the world X observatons g decson functon L[g,y] loss of predctng y wth g Bayes decson rule s
Lecture 2 Solution of Nonlinear Equations ( Root Finding Problems )
Lecture Soluton o Nonlnear Equatons Root Fndng Problems Dentons Classcaton o Methods Analytcal Solutons Graphcal Methods Numercal Methods Bracketng Methods Open Methods Convergence Notatons Root Fndng
Valuated Binary Tree: A New Approach in Study of Integers
Internatonal Journal of Scentfc Innovatve Mathematcal Research (IJSIMR) Volume 4, Issue 3, March 6, PP 63-67 ISS 347-37X (Prnt) & ISS 347-34 (Onlne) wwwarcournalsorg Valuated Bnary Tree: A ew Approach
The Dirac Equation for a One-electron atom. In this section we will derive the Dirac equation for a one-electron atom.
The Drac Equaton for a One-electron atom In ths secton we wll derve the Drac equaton for a one-electron atom. Accordng to Ensten the energy of a artcle wth rest mass m movng wth a velocty V s gven by E
On the Reducibility of Cyclotomic Polynomials over Finite Fields
On the Reducblty of Cyclotomc Polynomals over Fnte Felds Brett Harrson under the drecton of Mk Havlíčková Massachusetts Insttute of Technology Research Scence Insttute August 2, 2005 Abstract The rreducblty
THE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructions
THE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructons by George Hardgrove Chemstry Department St. Olaf College Northfeld, MN 55057 hardgrov@lars.acc.stolaf.edu Copyrght George
Discussion 11 Summary 11/20/2018
Dscusson 11 Summary 11/20/2018 1 Quz 8 1. Prove for any sets A, B that A = A B ff B A. Soluton: There are two drectons we need to prove: (a) A = A B B A, (b) B A A = A B. (a) Frst, we prove A = A B B A.
SOME RESULTS ON TRANSFORMATIONS GROUPS OF N-LINEAR CONNECTIONS IN THE 2-TANGENT BUNDLE
STUDIA UNIV. BABEŞ BOLYAI MATHEMATICA Volume LIII Number March 008 SOME RESULTS ON TRANSFORMATIONS GROUPS OF N-LINEAR CONNECTIONS IN THE -TANGENT BUNDLE GHEORGHE ATANASIU AND MONICA PURCARU Abstract. In
Air Age Equation Parameterized by Ventilation Grouped Time WU Wen-zhong
Appled Mechancs and Materals Submtted: 2014-05-07 ISSN: 1662-7482, Vols. 587-589, pp 449-452 Accepted: 2014-05-10 do:10.4028/www.scentfc.net/amm.587-589.449 Onlne: 2014-07-04 2014 Trans Tech Publcatons,
(2mn, m 2 n 2, m 2 + n 2 )
MATH 16T Homewk Solutons 1. Recall that a natural number n N s a perfect square f n = m f some m N. a) Let n = p α even f = 1,,..., k. be the prme factzaton of some n. Prove that n s a perfect square f
High-Order Hamilton s Principle and the Hamilton s Principle of High-Order Lagrangian Function
Commun. Theor. Phys. Bejng, Chna 49 008 pp. 97 30 c Chnese Physcal Socety Vol. 49, No., February 15, 008 Hgh-Orer Hamlton s Prncple an the Hamlton s Prncple of Hgh-Orer Lagrangan Functon ZHAO Hong-Xa an
EXACT TRAVELLING WAVE SOLUTIONS FOR THREE NONLINEAR EVOLUTION EQUATIONS BY A BERNOULLI SUB-ODE METHOD
www.arpapress.co/volues/vol16issue/ijrras_16 10.pdf EXACT TRAVELLING WAVE SOLUTIONS FOR THREE NONLINEAR EVOLUTION EQUATIONS BY A BERNOULLI SUB-ODE METHOD Chengbo Tan & Qnghua Feng * School of Scence, Shandong
829. An adaptive method for inertia force identification in cantilever under moving mass
89. An adaptve method for nerta force dentfcaton n cantlever under movng mass Qang Chen 1, Mnzhuo Wang, Hao Yan 3, Haonan Ye 4, Guola Yang 5 1,, 3, 4 Department of Control and System Engneerng, Nanng Unversty,
Homework 9 Solutions. 1. (Exercises from the book, 6 th edition, 6.6, 1-3.) Determine the number of distinct orderings of the letters given:
Homework 9 Solutons PROBLEM ONE 1 (Exercses from the book, th edton,, 1-) Determne the number of dstnct orderngs of the letters gven: (a) GUIDE Soluton: 5! (b) SCHOOL Soluton:! (c) SALESPERSONS Soluton:
Linear system of the Schrödinger equation Notes on Quantum Mechanics
Lnear sstem of the Schrödnger equaton Notes on Quantum Mechancs htt://quantum.bu.edu/notes/quantummechancs/lnearsstems.df Last udated Wednesda, October 9, 003 :0:08 Corght 003 Dan Dll (dan@bu.edu) Deartment
12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA. 4. Tensor product
12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA Here s an outlne of what I dd: (1) categorcal defnton (2) constructon (3) lst of basc propertes (4) dstrbutve property (5) rght exactness (6) localzaton
Integrals and Invariants of Euler-Lagrange Equations
Lecture 16 Integrals and Invarants of Euler-Lagrange Equatons ME 256 at the Indan Insttute of Scence, Bengaluru Varatonal Methods and Structural Optmzaton G. K. Ananthasuresh Professor, Mechancal Engneerng,
Implementation of the Matrix Method
Computatonal Photoncs, Prof. Thomas Pertsch, Abbe School of Photoncs, FSU Jena Computatonal Photoncs Semnar 0 Implementaton of the Matr Method calculaton of the transfer matr calculaton of reflecton and
J. Number Theory 130(2010), no. 4, SOME CURIOUS CONGRUENCES MODULO PRIMES
J. Number Theory 30(200, no. 4, 930 935. SOME CURIOUS CONGRUENCES MODULO PRIMES L-Lu Zhao and Zh-We Sun Department of Mathematcs, Nanjng Unversty Nanjng 20093, People s Republc of Chna zhaollu@gmal.com,
Restricted divisor sums
ACTA ARITHMETICA 02 2002) Restrcted dvsor sums by Kevn A Broughan Hamlton) Introducton There s a body of work n the lterature on varous restrcted sums of the number of dvsors of an nteger functon ncludng
Lecture 5.8 Flux Vector Splitting
Lecture 5.8 Flux Vector Splttng 1 Flux Vector Splttng The vector E n (5.7.) can be rewrtten as E = AU (5.8.1) (wth A as gven n (5.7.4) or (5.7.6) ) whenever, the equaton of state s of the separable form
First day August 1, Problems and Solutions
FOURTH INTERNATIONAL COMPETITION FOR UNIVERSITY STUDENTS IN MATHEMATICS July 30 August 4, 997, Plovdv, BULGARIA Frst day August, 997 Problems and Solutons Problem. Let {ε n } n= be a sequence of postve
Foundations of Arithmetic
Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an
Combined Wronskian solutions to the 2D Toda molecule equation
Combned Wronskan solutons to the 2D Toda molecule equaton Wen-Xu Ma Department of Mathematcs and Statstcs, Unversty of South Florda, Tampa, FL 33620-5700, USA Abstract By combnng two peces of b-drectonal
Mean Field / Variational Approximations
Mean Feld / Varatonal Appromatons resented by Jose Nuñez 0/24/05 Outlne Introducton Mean Feld Appromaton Structured Mean Feld Weghted Mean Feld Varatonal Methods Introducton roblem: We have dstrbuton but
Newton s Method for One - Dimensional Optimization - Theory
Numercal Methods Newton s Method for One - Dmensonal Optmzaton - Theory For more detals on ths topc Go to Clck on Keyword Clck on Newton s Method for One- Dmensonal Optmzaton You are free to Share to copy,
Similar Constructing Method for Solving the Boundary Value Problem of the Compound Kummer Equation
Amercan Journal of Mathematcal Analyss, 05, Vol. 3, No., 39-43 Avalable onlne at http://pubs.scepub.com/ajma/3//3 Scence and Educaton Publshng DOI:0.69/ajma-3--3 Smlar Constructng Method for Solvng the
Chapter 4 The Wave Equation
Chapter 4 The Wave Equaton Another classcal example of a hyperbolc PDE s a wave equaton. The wave equaton s a second-order lnear hyperbolc PDE that descrbes the propagaton of a varety of waves, such as
2 More examples with details
Physcs 129b Lecture 3 Caltech, 01/15/19 2 More examples wth detals 2.3 The permutaton group n = 4 S 4 contans 4! = 24 elements. One s the dentty e. Sx of them are exchange of two objects (, j) ( to j and
: Numerical Analysis Topic 2: Solution of Nonlinear Equations Lectures 5-11:
764: Numercal Analyss Topc : Soluton o Nonlnear Equatons Lectures 5-: UIN Malang Read Chapters 5 and 6 o the tetbook 764_Topc Lecture 5 Soluton o Nonlnear Equatons Root Fndng Problems Dentons Classcaton
= z 20 z n. (k 20) + 4 z k = 4
Problem Set #7 solutons 7.2.. (a Fnd the coeffcent of z k n (z + z 5 + z 6 + z 7 + 5, k 20. We use the known seres expanson ( n+l ( z l l z n below: (z + z 5 + z 6 + z 7 + 5 (z 5 ( + z + z 2 + z + 5 5
Rao IIT Academy/ SSC - Board Exam 2018 / Mathematics Code-A / QP + Solutions JEE MEDICAL-UG BOARDS KVPY NTSE OLYMPIADS SSC - BOARD
Rao IIT Academy/ SSC - Board Exam 08 / Mathematcs Code-A / QP + Solutons JEE MEDICAL-UG BOARDS KVPY NTSE OLYMPIADS SSC - BOARD - 08 Date: 0.03.08 MATHEMATICS - PAPER- - SOLUTIONS Q. Attempt any FIVE of
ALGEBRA HW 7 CLAY SHONKWILER
ALGEBRA HW 7 CLAY SHONKWILER 1 Whch of the followng rngs R are dscrete valuaton rngs? For those that are, fnd the fracton feld K = frac R, the resdue feld k = R/m (where m) s the maxmal deal), and a unformzer
Quadratic Formula, Completing the Square, Systems Review Sheet
Quadratc Formula Completng the Square Systems Revew Sheet 1. Factor the polynomal completely. 6. Use the graph to approxmate the real zeros of the functon. 2. Fnd the real-number solutons of the equaton.
Math 2534 Final Exam Review Answer Key 1. Know deþnitions for the various terms: a. Modus Ponens b. Modus Tollens c. divides d. rational e.
Math 534 Fnal Exam Revew Answer Key 1. Know deþntons for the varous terms: a. Modus Ponens b. Modus Tollens c. dvdes d. ratonal e. Quotent-Remander Theorem f. unon g. ntersecton h. complement. DeMorgan
Non-Ideality Through Fugacity and Activity
Non-Idealty Through Fugacty and Actvty S. Patel Deartment of Chemstry and Bochemstry, Unversty of Delaware, Newark, Delaware 19716, USA Corresondng author. E-mal: saatel@udel.edu 1 I. FUGACITY In ths dscusson,
Chapter 2 Transformations and Expectations. , and define f
Revew for the prevous lecture Defnton: support set of a ranom varable, the monotone functon; Theorem: How to obtan a cf, pf (or pmf) of functons of a ranom varable; Eamples: several eamples Chapter Transformatons
n α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
risk and uncertainty assessment
Optmal forecastng of atmospherc qualty n ndustral regons: rsk and uncertanty assessment Vladmr Penenko Insttute of Computatonal Mathematcs and Mathematcal Geophyscs SD RAS Goal Development of theoretcal
Chowla s Problem on the Non-Vanishing of Certain Infinite Series and Related Questions
Proc. Int. Conf. Number Theory and Dscrete Geometry No. 4, 2007, pp. 7 79. Chowla s Problem on the Non-Vanshng of Certan Infnte Seres and Related Questons N. Saradha School of Mathematcs, Tata Insttute
MAE140 - Linear Circuits - Fall 13 Midterm, October 31
Instructons ME140 - Lnear Crcuts - Fall 13 Mdterm, October 31 () Ths exam s open book. You may use whatever wrtten materals you choose, ncludng your class notes and textbook. You may use a hand calculator