The Relationship of the Cotangent Function to Special Relativity Theory, Silver Means, p-cycles, and Chaos Theory

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Randolf Abel Willis
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1 Origial Paper Forma, 8, 49 6, 003 The Relatioship of the Cotaget Futio to Speial Relativity Theory, Silver Meas, p-yles, ad Chaos Theory Jay KAPPRAFF * ad Gary W ADAMSON New Jersey Istitute of Tehology, Newark, NJ 070, USA PO Box 457, Sa Diego, CA 9-457, USA * address: kappraff@aolom (Reeived Otober 4, 003; Aepted November 8, 003) Keywords: Cotaget, Relativity, Silver Meas, Periodi Trajetories, Chaos Theory Abstrat The otaget futio is show to play a key role i speial relativity theory, silver mea ostats, p-yles for all positive iteger values of p, ad period doublig bifuratios from haos theory Geometry ad properties of umber play importat roles i the aalysis A ew theorem pertaiig to periodi otiued fratios is itrodued Itrodutio The otaget futio has bee foud to play a role i speial relativity theory where the stadard formulas are expressed i terms of the multiplier that determies the magitude of the Doppler effet rather tha the usual v/ where v is the veloity of a partile ad is the speed of light Also a sequee of silver mea ostats that arise aturally i dyamial systems theory are haraterized by the otaget The otaget is foud to haraterize a map that has periods of all legths Fially, the otaget futio is a atural way to express the values of the dyamial variable approahig haos at a poit of period doublig bifuratio Speial Relativity ad the Cotaget Futio Aordig to speial relativity theory, a body movig at veloity v experiees a shorteig i legth L by the fator L = L/β, a dilatio of time t by t = tβ ad a irease i mass m by the fator m = mβ where β is the Lorez oeffiiet, β = v / () Also if two partiles are movig o a straight lie with respet to eah other, the frequey 49

2 50 J KAPPRAFF ad G W ADAMSON ω of the light that they emit varies aordig to ω = ω/k while the wavelegth of that light varies by λ = λk This is kow as the Doppler effet ad we refer to k as the Doppler oeffiiet Whe the partiles move i opposite diretios the k > ad the frequey shift is towards the red whereas whe the partiles are movig towards eah other 0 < k < ad the shift is towards the blue where, v + k = v / / ( ) Geerally v/ is speified ad all alulatios follow We osider the Doppler oeffiiet k to be the parameter with whih to referee all other fators If we set k = otθ for 0 θ π/, the, It follows algebraially that ad ot θ = k sih l 3 k = k = k () k β = osh l k ( 4) v/ = tah l k ( 5) This a be visualized both i terms of the geometry of a irle ad that of a hyperbola First osider the triagle show i Fig From this, k os θ = / 6 + = v k ( ) Fig A right triagle defies osθ, siθ, ad otθ i terms of the parameter k

3 The Relatioship of the Cotaget Futio to Speial Relativity Theory 5 k si θ = / β 7 k + = ( ) I Fig these relatioships are pitured i a irle Settig x = oshu ad y = sihu where u = l k these relatioships a be pitured o a hyperbola as show i Fig 3 where the area subteded by the urve ad rays emaatig from the origi is A = u/ = l k The fat that the veloity is limited by the speed of light is refleted i the urve of tah lk show i Fig 4 where v as k ad v as k 0 It is well kow that whe veloities are added relativistially their sum is v v v + = v v + Fig The parameters v/ ad β are represeted by the geometry of a irle Fig 3 The parameters β ad k are represeted by the geometry of a hyperbola

4 5 J KAPPRAFF ad G W ADAMSON Fig 4 The futio v/ = tah lk illustrates the speed of light as the limitig veloity of a partile Adamso has geeralized this formula to a arbitrary umber of veloities, v 3 Silver Meas ad the Cotaget Futio The solutio to = v v v + + tah tah tah tah + L 3 8 x = N for positive iteger N ( 9) x is the N-th silver mea of the first kid, SM (N) whih arises aturally i dyamial systems theory (SCHROEDER, 990) If N = the x = SM () = (+ 5 )/, the golde mea, where as if N =, the x = SM () = + referred to as the silver mea As a result of Eq (3) it follows that, () N sih SM N e ( )= ( 0) Also if k = SM (N) the v/ = N/ 4 + N ad β = 4 + N / follows from Eqs (3), (4), (5), ad (9) 4 p-yles of the Cotaget Map Cosider the map, x sih lx (a)

5 whih has the property that The Relatioship of the Cotaget Futio to Speial Relativity Theory 53 otθ otθ (b) where ow x = otθ Sie sih lx = (x )/x it follows that this map is derived by applyig Newto s method, z z f(z)/f (z) for z omplex, to the futio f(z) = z + I other words it is the map that loates the root equal to the imagiary umber i If a seed x 0 = otπ/ for a odd iteger is take i map a, the the trajetory is x 0, x, x,, x j, where x j = ot(π j /) as a result of Eq (b) The trajetory is osidered to be periodi with period p if x 0 = x p where p is the smallest iteger for whih j = (mod ) The trajetory is said to have a p-yle Likewise, the map otθ otmθ has p-yles whe m p (mod ) () for m relatively prime to This mappig a also be arried out by ratioal futios ot show here For prime may p-values were tabulated by Beiler (BEILER, 964) If is a arbitrary odd iteger the values of p are the result of the followig Theorems Theorem (Euler Fermat Theorem): If m ad are positive itegers ad m ad are relatively prime, the m φ() (mod ) where φ() is the Euler phi futio (APOSTOL, 976) Remark: If is prime the φ() = ad Theorem redues to Fermat s Little theorem Corollary : If m ad are relatively prime the m j for j = 0,,, 3, forms a p-yle where p is a fator of φ() Proof: The Corollary is a immediate osequee of Theorem ad Eq () Defiitio: The iteger m is a primitive root modulus if ad oly if for arbitrary k (mod ) relatively prime to there exists a uique iteger j suh that m j = k (mod ) where j is deoted by j = id m k (SCHROEDER, 990) Theorem : If m is a primitive root modulus the p = φ() Proof: If m is a primitive root modulus, the j = id m k for eah of the φ() values of k relatively prime to Theorem 3: If m is ot a primitive root modulus the there is more tha oe p-yle The umber T of p-yles is give by T = φ()/p How are these T, p-yles foud? The olletio of poits {x k = ot(πk/)} for k relatively prime to are iluded i the T, p-yles orrespodig to Let S be the set of idies relatively prime to for k < φ(), Trajetory has idies, S = {k 0, k,, k,, k φ() } () j () j p k m : k, m, m,, m,, m mod j = = = ( ) 0

6 54 J KAPPRAFF ad G W ADAMSON orrespodig to the p-vetor, ( 0 j p ) k () k () k () k () () = =,,,,, k Let k 0 (i) be the smallest iteger from S ot used i trajetories,,, i The Trajetory i has idies, () i i j () i () i () i () i j () i p () i k k m : k, k m, k m,, k m,, k m k mod j = = ( ) orrespodig to the p-vetor k r () i = (k (i) 0, k (i),, k (i) j,, k (i) p ) The proess otiues util i = T at whih poit all idies from S have bee exhausted These theorems are illustrated by the followig three examples for the ase of m = Example : = where is a primitive root modulo S = {,, 3, 4, 5, 6, 7, 8, 9, 0} () k j = j :,, 4, 8, 6 = 5, 3 = 0, 64 = 9, 8 = 7, 56 = 3, 5 = 6, 04 = (mod ) so that p = 0 ad T = where k () = (,, 4, 8, 5, 0, 9, 7, 3, 6) Example : = 7 where is ot a primitive root modulus 7 S = {,, 3, 4, 5, 6} () k j = j :,, 4, 8 = (mod 7) so that p = 3 ad T = where k () () = (,, 4) Therefore k 0 () = 3 ad there is a seod 3-yle give by k j = 3 j : 3, 6, = 5, 4 = 3 (mod 7) where k ( ) = (3, 6, 5) Example 3: = 9 where is a primitive root modulus 9 S = {,, 4, 5, 7, 8} () k j = j :,, 4, 8, 6 = 7, 3 = 5, 64 = (mod 9) so that p = 6 ad T = where k () = (,, 4, 8, 7, 5) Depedig o, there exist p-yles of all legths related to the doublig of the argumet of the otaget futio Aother sequee of p-yles was show to be related to the doublig of the argumet of the osie futio These were show to orrespod to the Madelbrot map at the extreme left had poit o the real axis, a poit of haos (KAPPRAFF, 00; KAPPRAFF ad ADAMSON, 004) 5 The Cotaget Futio ad Chaos Theory Steve Strogatz has desribed the followig geeral theory of period doublig bifuratios (STROGATZ, 994) Let f(x, µ) be ay uimodal map that udergoes a perioddoublig route to haos Suppose that the variables are defied suh that the period- yle is bor at x = 0 whe µ = 0 The for both x ad µ lose to 0, the map is approximated by ( ) + + ( ) x+ = + µ x ax L 3a sie the eigevalue is at the bifuratio If the variable x is resaled by x x/a, the Eq (3a) a be rewritte as,

7 The Relatioship of the Cotaget Futio to Speial Relativity Theory 55 ( ) + + ( ) x+ = + µ x x L 3b By defiitio of period-, p is mapped to q ad q to p Therefore Eq (3b) yields, p = ( + µ)q + q ad q = ( + µ)p + p By subtratig oe of these equatios from the other, ad fatorig out p q, ad the multiplyig these two equatios together ad simplifyig yields p + q = µ (4a) ad pq = µ (4b) From whih it follows that µ + µ + 4µ p = ( 5a) ad µ µ + 4µ q = 5b ( ) Sie the map x f (x) has the same form as Eq (3) it otiues to hold as p ad q bifurate Fig 5 The geometry of a right triagle represets the trajetory values p ad q after a period doublig bifuratio where the radius of the isribed irle r = q, lie segmet AM = p, ad the area of the retagle formed by r ad AM equals µ

8 56 J KAPPRAFF ad G W ADAMSON We have foud a simple geometri represetatio of Eq (5), oe agai usig the otaget futio Cosider the right triagle ABC i Fig 5 with sides a, b, ad The area of triagle ABC is omputed by the equatio, A triagle = sr (6) where s is the semi-perimeter ad r the radius of the isribed irle From Eq (6) it follows that, From Fig 5 it is lear that, r ab 7 a b = + + ( ) A b r ot = ( 8) r Usig Eqs (6) ad (7) it follows after some algebra that ot(a/) satisfies the equatio, x x a + = ( 9) where is the legth of the hypotheuse of the triagle ABC The solutio x > to this equatio is referred to as the silver mea of the seod kid SM whe a = ad is a iteger, ie, the solutio to x + (/x) = is x = ( )= + 4 SM for x > ( 0a) By the symmetry of this equatio the other solutio is 4 x = SM Let a = ad = N + i whih ase we have show that, Similar to Eq (0), ( ) = ( 0b) A ot = SM N + ( ) ( )

9 The Relatioship of the Cotaget Futio to Speial Relativity Theory 57 N osh SM N e ( )= ( ) Usig Eqs (7) ad (8) with a =, it is a exerise i algebra to show that, ad ( ) ( ) p = SM N + 3a q = r ( N + ) = ( 3b SM ) where µ = whih equals N whe = N + This is see geometrially i Fig 5 where the retagle with sides r ad AM has area, A ret = r(b r) = N From Eq (4a), p + q = N whe µ = N a iteger However, it a be easily show that ad + = ( ) ( 4a) p q f N p + q p q = g ( N) ( 4b) for a positive iteger ad f (N) ad g (N) iteger futios of N where f (N) = N, N(N + ), ad (N + ) (N ) + 4 for =,, ad 3 respetively ad g (N) = ad N for = ad respetively Equatio (4a) is prove by replaig p ad q i Eq (4) by their values from Eqs (3) ad usig the fat that SM ( N + )+ SM N ( N + ) = + Equatio (4b) is prove by replaig p ad q i Eq (4) by their values i Eq (3) ad usig Eq (0) with = N + Equatio (4) are geeralized forms of Biet s formula give by, α α + β + β = L ( N) ad = F ( N) α β where α = SM (N) ad β = {/[SM (N)]}, both satisfyig Eq (9), L (N) are geeralized Luas umbers with L () =, 3, 4, 7,, for =,, 3,, ad F (N) are geeralized Fiboai umbers with F () =,,, 3, 5, for =,, 3,

10 58 J KAPPRAFF ad G W ADAMSON Also it a be show that, A r B r C r ta =, ta =, ad = ( 5) s a s b s The silver meas also have a lose oetio to otiued fratios with periodi idies as show i Appedix Based o this oetio we state the followig theorem the proof of whih is give i Appedix Theorem: For a ad b positive itegers, if [a, b, a, b, ] = [ ab, ] is a otiued fratio with repeated idies of period, the If a = ad b = N the this yields, [ ab, ] b [ ba, ] = a ad [ ba, ] [ ]= ab, ab [, N] N [ N, ] = ad [ N, ] [ ]=, N N It follows from Eq (4) that p ad q for µ = N a iteger a be expressed as the otiued fratios, p = [ N,, N,, N,, K]= [ N, ] ad q = [, N,, N,, N, K]= [, N] ( 6) The otiued fratio expasios of p ad q for µ = N a be verified by the omputatioal proedure give i Appedix We have foud aother appliatio of Eqs (5a) ad (5b) Shroeder has desribed the harateristi impedae of the ifiite eletrial ladder iruit show i Fig 6 (SCHROEDER, 990) The impedae Z 0 is give by the formula, 4R Z0 = R+ R + 7 G, ( ) where R ad G are the resistaes of the iruit However if we let Z 0 = /r, where r = q = µ, the sie q = µ/p, from Eq (4b) we have, Z 0 4 = + + µ

11 The Relatioship of the Cotaget Futio to Speial Relativity Theory 59 Fig 6 A ifiite ladder iruit with resistaes R ad G This orrespods to a ormalized iruit with R = ad G = µ i Eq (7) If µ = N, a iteger, the usig Eq (6), the followig oise formula is obtaied, 6 Colusio Z 0 = [, N ] The otaget futio has bee show to spa the areas of speial relativity theory ad the silver meas of the first ad seod kid It is useful for dyamial systems theory, leadig to a haraterizatio of periodi orbits of all legths, ad formulas goverig the period doublig bifuratios of haos theory The period doublig bifuratios are see to be diretly oeted to silver meas ad lead to a geeralizatio of Biet s formula for expressig Fiboai umbers The otaget futio also arises i eletrial iruit theory It is sigifiat that these oetios are losely related to both geometry ad the theory of umbers Appedix Ay real umber a be expressed as a otiued fratio of the form, α = a + a + a3 + = [ a; aa3ka k K] where the idies a, a, a 3, are positive itegers If α is a umber betwee 0 ad the the leadig iteger a be elimiated If the idies repeat with period p the this is idiated by the followig otatio, α = [ aa Kaaaa p 3Kaa p K]= [ aaa 3K ap] Cosider a real umber 0 < α < expressible as a otiued fratio with periodi idies [ 3 Kk Kp p] The approximats obtaied by truatig the otiued

12 60 J KAPPRAFF ad G W ADAMSON fratio at the k-th positio is give by a k /b k for k =,, 3,, p, ie, where a b 3 a a a a a a 3 k p b b b b b b 3 k p p k p a a3 a 3 + a ak a k k+ + ak =, =, =, = for k 3 b + b b + b b b + b Cosider the matrix, 3 3 k p p k k+ k M = a b d where a = a p, b = b p, = a p, ad d = b p We a ow state the followig theorem Theorem: Let a/b ad /d be the (p )-th ad p-th approximat to the otiued fratio expasio of α where 0 < α < ad has periodi idies of period p If M = a b d, [ ] = /(λ a) where λ = SM (TrM) for p odd ad λ = SM (TrM) the α = Kk Kp p for p eve Proof: We outlie the proof with details left to the reader Cosider iterates of a periodi otiued fratio K K p [ k p p] with period ( )+ a a ap ap ap ap a ( )+ b b b b b b b p p p p p p p p ( )+ p p p where we let a = a p, b = b p, = a p, ad d = b p Let M = a b d

13 The Relatioship of the Cotaget Futio to Speial Relativity Theory 6 It a be show by idutio that a b p p d p p = M Therefore ap [ 3Kk Kp p]= lim b The harateristi equatio of M is give by: det(m λi) = λ λtrm + detm But TrM = a + d ad for otiued fratio expasios it is well kow that det M = whe p is eve ad whe p is odd Therefore the eigevalues satisfy the equatio, λ ± /λ = TrM = a + d = t ad λ = SM (t) for p eve ad λ = SM (t) for p odd It follows that p 4 = t + t m λ ad that λ = /λ for p eve ad λ = (/λ ) for p odd where λ > ad λ < Sie there is a omplete set of eigevalues, M is diagoalizable Diagoalizig M, it follows that M = V V λ 0 0 λ where V is a matrix of eigevetors of M Lettig λ = λ ad reogizig that λ = ±(/λ ), λ V = λ a ± λa ad V = ( ) λ ± λa a λ λ where the positive sig orrespods to p odd ad the egative sig to p eve After a legthy omputatio, we fid that a p + = λ ( ± ) λ + λa ad b = λ ( λ a)+ ± λ p ( ) ( )

14 6 J KAPPRAFF ad G W ADAMSON Sie λ >, it follows that, ap [ 3Kk Kp p]= lim = b λ a p Corollary: [ ab, ] b ad [ ba, ] = a [ ba, ] [ ]= ab, ab Proof: For [ ab, ] it follows that M = a b ab+ ad therefore, ab, [ ] = b/(λ ) Likewise it a be show that for ba, [ ], M = b a ab+ [ ] = a/(λ ) Sie λ = λ the first part of the orollary follows The seod part ad ba, follows i a similar maer makig use of the fat that λ =SM (ab + ) REFERENCES APOSTOL, T M (976) Itrodutio to Aalyti Number Theory, Spriger-Verlag, New York BEILER, A H (964) Rereatios i the Theory of Numbers, Dover, New York KAPPRAFF, J (00) Beyod Measure: A Guided Tour through Nature, Myth, ad Number, World Sietifi, Sigapore KAPPRAFF, J ad ADAMSON, G W (004) Polygos ad haos, Jour of Dyam Syst with Geom Theories (i press) KOSHY, T (00) Fiboai ad Luas Numbers with Appliatios, Wiley, New York SCHROEDER, M (990) Fratals, Chaos, Power Law, WH Freema Press, New York STROGATZ, S H (994) Noliear Dyamis ad Chaos, Perseus Books, Cambridge, MA

Bernoulli Numbers. n(n+1) = n(n+1)(2n+1) = n(n 1) 2

Bernoulli Numbers. n(n+1) = n(n+1)(2n+1) = n(n 1) 2 Beroulli Numbers Beroulli umbers are amed after the great Swiss mathematiia Jaob Beroulli5-705 who used these umbers i the power-sum problem. The power-sum problem is to fid a formula for the sum of the