Komentary on W W Sawyer's A Path To Modern Mathematics ( 1966 ).

Disklaimer: The folowing thry web pajes wer riten by an out-sider to mathematiks. They hav not byn independently chek'd. They ar only a personal atempt at understanding, tho houpfuly helpful to others.

( 1 ) Fibonacci serys and the damp'd wav equation.

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( Notation:
subskripts to a term ar shown by a koma in betwyn the term and the symbols usualy subskripted: x,n myns the n-th term of x insted of the konvention that n is a sub-skript.
Super-skripts, such as powers, ar shown by a quotation mark betwen the term and its power: x'n myns x to the power of n. But the usual notation x is yus'd for x squar'd.
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Fibonacci serys.

The Fibonacci serys okur in natural forms, such as the aranjment of pIn-koun or pInapl worls or sun-flower petals. It has byn wIdly study'd by mathematicians.

Solving the formula for the Fibonacci serys is perhaps as ysy a way as any to understand sertain equations of wId-spred yus in siens. The serys starts with a pair of numbers, most simply 0 and 1. They ar aded, to produs the next number in the serys, wich is aded, in turn, to the previus number, and so on:

0 + 1 = 1; 1 + 1 = 2; 1 + 2 = 3; 2 + 3 = 5; 3 + 5 = 8; etc.

That is, the Fibonacci serys gos: 0, 1, 1, 2, 3, 5, 8, 13, 21, etc.

A graf kan be drawn of the aded pairs, as the x and y axes, respektivly. The x valus ko-insId with the houl serys, and the y valus with al but the first term, zero.

Chanjing the ko-ordinats to the graf of the Fibonacci serys.

The progres of the red zig-zag lIn, wich marks the suksesiv pairs of the serys, is direktly mesur'd by the gryn ko-ordinats, as such, the so-kal'd eigen-vektors.

Finding the formula for the serys amounts to this: given any first pair of numbers, a,0 and a,1, ( most simply 0 and 1 ) wat is the valu of the n-th term, a,n? Hyr, a,n kan myn the tenth, nIntynth, seventy-first, or any other order of term in the serys. It ko-insIds with the n-th term of the x ordinat, x,n. In other words, a,n = x,n.

( The normal notation, for the order of term, folows serys leter a, or x, with a sub-skript number or sub-skript leter n. )

The graf's red lIn shows the suksesiv pairs of the serys. They get kloser and kloser to the X-axis of a nw ko-ordinat system ( drawn in gryn ). This nw ordinat konveniently shows the limit to wich the serys aproches for larjer and larjer valus of the pairs in ratio, y/x: 1/1, 2/1, 3/2, 5/3, 8/5, etc. This limit is 1/2( 1 + 5 ) or about 1.618. The X-ordinat has the equation y ~ 1.618x, in the orijinal ko-ordinats.

The other nw ordinat, Y, has the equation y = - 0.618x. The negativ sIn is with respekt to the fakt that, with ych step of the serys, the red zig-zag skips from positiv to negativ sIds of the nw, Y-axis. This also alows the zig-zag to be trais'd bak ( sy the doted red lIn on the graf ) to - 0.618 on the Y-axis.
Taking the diminishing zig zag as the krudest representation of a damp'd wav or lesening osilation, this is its starting point lyding direktly to the orijinal maximaly displas'd amplitud of the wav, befor its enerjy is gradualy lost in the damping down, as by a shok absorber.


The karakteristik equation for eigen-funktions and eigen-valus.

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Finding the limiting valu of the serys is establish'd by a standard prosedur, involving the 'karakteristik equation': T - ( a + d )T + ( ad - bc)I = 0. ( Text-bwks komonly suply a prwf of this equation. )

Simply substitut the first tw pair numbers, 0, 1 and 1, 1, of the serys into a, b, c, d, respektivly: T - ( 0 + 1 )T + ( 0x1 - 1x1 )I = 0. And yus the formula for solving quadratic equations ( wer the leters a, b, c, ar put to a diferent yus: a is the ko-eficient of T, b is ko-eficient of T, c is ko-eficient of I, for Identity, wich is esentialy just a unitary term ). So:

T = ( - b ( b - 4ac ))/2a. = 1/2( 1 5 ) ~ 1.618 or - 0.618.

( Notis that we nyded tw distinkt rwts for tw nw ko-ordinats. )

The T is for transformation that tels how much ych member of a pair gets simply expanded or shrunk, in a strait lIn, with the next step of the serys, in this nw ko-ordinat system. In other words, if X, Y ar a pair and X*, Y* ar ther next steps or suksyding transformations, then X* ~ 1.618X and Y* ~ - 0.618Y. Ko-ordinats wich hav this simpl linear property ar kal'd 'eigen-vektors'. And ther magnituds, or the amounts they get expanded or kontrakted, ar kal'd ther 'eigen-valus'.
( Thys sentral terms of 'linear analysis', the uons 'modern' mathematiks, of wich Sawyer spok, haunt texts of mathematiks and modern fysiks. )

In the nw ko-ordinats, the first pair of the Fibonacci serys, X,0 and Y,0, may be found as folows, with referens to the abov graf. The tanjent of angl B is in the ratio of 1 over 1.618, or, tan B = 1/1.618. Therfor, angl B = 31.718 degres. Subtrakt from 90 degres to find angl A = 58.232 degres.
SIn A = X,0/1 = 0.85.
Kos A = Y,0/1 = 0.526.

Al the suksyding valus of X and Y may be found, respektivly, by multiplying thys initial valus by the nw ko-ordinats' eigen-valus, to the power of n ( hyr signify'd 'n insted of with index or super-skript n ) wer n is the order of the term in the serys being sot. In other words, X,n = X,0.( 1.618 )'n. And Y,n = Y,0.( - 0.618 )'n.

Sy tabl 1 ( komparing with the graf of both ko-ordinat systems ):

Tabl 1: Fibonacci serys' x-y pairs, konverted into ther X-Y ( eigen-funktion ) pairs, for any n-th order pair. The first six pairs ar valu'd.
n x,n X,n y,n Y,n
0 0 0.85 1 0.526
1 1 1.375 1 - 0.325
2 1 2.225 2 0.201
3 2 3.600 3 - 0.124
4 3 5.825 5 0.077
5 5 9.426 8 - 0.047

If the diminishing zig-zag wer konsider'd as a simplify'd model of a damp'd wav or osilation petering out, the X-axis is the equilibrium lIn to wich the disturbanses finaly setl. Wer-as the Y-axis mesurs the amplituds of the vibrations as ther krests and trofs, about either sId of the X-axis, get smaler and smaler.


Fibonacci serys formula.

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Tabl 2 shows how to konvert the n-th term bak into an orijinal x-axis ryding, x,n. That is any term in the Fibonacci serys:
x,n = C( 1.618 )'n + D( - 0.618 )'n, wer C and D ar konstants.

This is the form of a solution to a sekond order equation. Indyd, the damp'd wav equation is a sekond order diferential equation.

The konstants ar determin'd by the first tw x terms, 'the initial konditions', of the serys, hyr zero and uon.

Taking x,n as zero,
0 = C( 1.618 )'0 + D( - 0.618 )'0 = C + D.
So, C = - D.

Taking x,n as uon,
1 = C( 1.618 )'1 + D( - 0.618)'1.

Substituting C for - D,
1 = C( 2.236 ). So, C = 0.447. And D = - 0.447.

Therfor, x,n = ( ( 1.618 )'n - ( - 0.618 )'n )0.447.
This is the Fibonacci serys formula, that we hav byn lwking for. See tabl 2 for exampls.

Tabl 2: Valuation for an n-th term in the Fibonacci serys, shown for the first six terms, ploted on the orijinal x-axis ( to tw desimal plases ).
( On the top row, the notation, 'n, replases the index letter n, for myning "to the power of n" ).
n ( 1.618 )'n - ( -0.618 )'n = Times 0.447;
x-values.
0 1 - 1 0 0
1 1.618 + 0.618 2.236 1
2 2.618 - 0.382 2.236 1
3 4.236 + 0.236 4.472 2
4 6.854 - 0.146 6.708 3
5 11.089 + 0.090 11.179 5


The karakteristik equation and the auxiliary equation.

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The Fibonacci serys is a diferens equation, say, x,n+1 - x,n = x,n-1. For instans, from the abov tabl of x-valus: 1 - 1 = 0, 2 - 1 = 1, 3 - 2 = 1, 5 - 3 = 2, etc.

Supos thys ar diferenses betwyn distanses ych kover'd in unit tIm, h = 1. ( How klosly the diferens equation aproximats the diferential equation depends on how fInly the tIm is mesur'd. But h mIt myn an interval of spas. For instans, wav krests may be a spatial funktion of the uniform wav-lenths betwyn them, as wel as forming periodikly in tIm. Insted of h = 1 hour, mesurment over minuts, h = 1/60, is mor akurat, as mesuring over sekonds, h = 1/360, is mor akurat stil. But this is not the konsern hyr. )

So , diferenses in distans, x,n+1 - x,n, and x,n - x,n-1, kover'd in tIm, h, ar ( x,n+1 - x,n )/h and ( x,n - x,n-1 )/h. This is nown as first order chanj, lIk velosity, as chanj in plas over tIm.

A sekond order chanj is a chanj in a chanj of valus, lIk akseleration, as a chanj in a chanj of plas over tIm, or a chanj in velosity over tIm. This mIt be ( 5 - 3 ) - ( 3 - 2) = 1.
Ych diferens ( x,n+1 - x,n )/h, ( x,n - x,n-1 )/h is, in turn, subtrakted, and divided by h, again, resulting in ( x,n+1 - 2x,n + x,n-1 )/h. A sekond order term, lIk this, is the hIest order term in a sekond order equation. Such an equation mIt be mayd up as folows. ( To simplify kalkulation, the tIm is unitary ).

With the nesesary ajustments to ther ko-eficients, a sekond order diferens is equated to a first order diferens and the n-th term, x,n, itself ( a zero order diferens ). Such a sekond order diferens equation is:

( x,n+1 + 2x,n - x,n-1 ) - 2( x,n - x,n-1 ) - x,n = 0.

This equation is esentialy the Fibonacci diferens equation, with the Fibonacci serys as the x ordinat valus, wer x is a funktion of n, the number of terms rych'd. The equation reduses to:

x,n+1 - x,n - x,n-1 = 0.

If this wer a sekond order diferential equation, its ko-eficients ( hyr, 1, 1, and 1 ) wud be adopted by an auxiliary equation, solv'd as a quadratik equation, just as the karakteristik equation was solv'd ( abov ). Hyr, the karakteristik equation is the way to a short kut or algorithm for finding the n-th term, insted of suksesivly working out a given serys terms ( normaly don by komputer ) til ryching the n-th term.

Diferens equations ar finit aproximations to diferential equations, by turning a problem in kalkulus to a problem in aljebra.


The transformation, T was found to be 1.618, the limiting ratio of y,n/x,n; or - 0.618, the limiting ratio of y,n/-x,n. This was lIk solving the sekond order equation ( of dependent variabl, y,n, to independent variabl, x,n ) as y = 1.618x or - 0.618x. Thys ar the linear equations of the eigen-funktions, X and Y, ( hws eigen-valus ar ) in terms of the orijinal ko-ordinats.


GId to kalkulating a diferential equation.

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The sekond order diferential equation, nown as the damp'd wav equation, is a komon-plas in text-bwks. Enjinering math bwks usualy giv work'd exampls, wich is wer I'v taken the folowing kais of a fors on a weit hws osilations under-go slIt damping, du to resistans from its imersion in a viskus fluid.

Without going into the details, the esential equation is:

dy/dt + 2.8dy/dt + 25y = 0.

The mid term is the resistans term, bekaus resistans is asum'd proportional to the velosity of the body. If this first order term wer omited, a simpl undamp'd wav equation wud be left. The ko-eficient of this first derivativ is low kompar'd to the ko-eficient of the y term, implIing only slIt damping of the osilating body in question.

The dependent variabl, y, is the displasment, at a given tIm, t, the independent variabl. This displasment mIt be of a weited spring, made to osilat by a fors puling upon it. By Newton's laws the fors is proportional to the mas tIms akseleration. Akseleration is the sekond derivativ of displasment with respekt to tIm ( the first term in the equation ). And the fors is equal and oposit to the displasment, wich is proportional to the elastisity of the spring.
Or, the faktor involv'd mIt be the tension of a vibrating string, if the displasment of a string is in question. ( The mas and elastik konstants dont show direktly in the abov equation, bekaus they ar kombin'd into the ko-eficient to the third term. )

As tIm gos on the osilations, of the wav from krest to trof, wil get smaler, as the wav dIs away. The prosedur for solving the valu of y, is to konsider the diferential equation in terms of an auxiliary equation:

m + 2.8m + 25 = 0.

The formula for solving quadratik equations ( quoted in an abov sektion ) givs: m = - 1.4 4.8i. For the kais of slIt damping ( wen b is les than 4ac in the quadratik formula ) the solution involvs komplex numbers.

This respektiv korespondens, of the the m terms to y and its first and sekond derivativs, is to do with the special property of a solution of y that kan be put in terms of the exponent, a konstant as important as pi. The exponent, given the symbol e, or 'exp' for short, is an infinit number of about 2.718...
This exponent is the sum of an infinit serys of terms. The exponent to the power of x, or exp(x) = x/0! + x/1! + x/2! + x/3! + ...

The first term always equals uon no mater the valu of x. If x equals uon, then the sum of the serys is 2.718... ( to thry desimal plases ).

The derivativ of ych term equals the term befor it, in the serys. So, taking the derivativ of the houl serys, the exponential funktion, efektivly lyvs it unchanj'd -- no mater how many tIms uon repyts the prosedur, taking derivativs of derivativs. The exponential funktion remains unchanj'd under diferentiations.

Jeneraly, the derivativs of a funktion giv a rait of chanj. Diferential kalkulus mesurs growth. Exponential growth is the special kais of a konstant rait. Provided, say, y = A.exp(mt), wer A and m ar konstants, the derivativ of y lyvs y esentialy unchanj'd. The only diferens that the first derivativ brings is multiplikation by m: dy/dt = Am.exp(mt). The sekond derivativ repyts the proses: dy/dt = Am.exp(mt). Wer A and B ar konstants.

With luk, the problem of solving the sekond order equation ( in terms of y ) reduses to solving a quadratik equation ( in terms of m ).

Hens, in our exampl, y = A.exp(- 1.4+i4.8) + B.exp(- 1.4-i4.8).

For equations, mathematicians defIn as 'linear', the tw solutions, in terms of m, kan be aded for a jeneral solution.

Another way of expresing the solution for y is:

y = exp(-1.4t)(C cos 4.8t + D sin 4.8t). Wer C and D ar konstants.

To solv numerikly for y, the valus of C and D may be found if the initial konditions ar nown, such as of y at t = 0. In this text-bwk exampl, y = - .4 at t = 0, so C = - .4.
In other words, at the start, t = 0, the diplasment is push'd from its equilibrium position at y = 0, to - .4. Subsequently, the displasment bounses bak in the positiv direktion to its bigest osilation, folow'd, in tIm, by lesening osilations, bak and forth akros the equilibrium lIn, til koming to rest again.

This exampl, modify'd from a text bwk, also gav another initial kondition, the velosity, at 16 fyt per sekond, of the osilating body, at t = 0, to asertain the other konstant, D.

The produkt rul of diferentiation, for y with respekt to t, is yus'd on the rIt hand sId of the equation for y, render'd as dy/dt, and set equal to sixtyn. Ther turn out to be only tw non-zero terms on the left hand sId, under diferentiation.

Hens, 16 = (-1.4 )( -.4 ) + 4.8D.

So, D is 3.22 ( to 2 desimal plases ).

Aktual valus of the displasment, y, kan now be found, at suksesiv intervals of tIm, starting at t = 0, wen y = - .4. For instans, after ( t = ) 0.29 sekonds, the body has about rych'd its maximum amplitud of osilation, or first and hIest krest, at ( y = ) 2.06 fyt ( to 2 desimal plases ).


GId to kalkulating a koresponding diferens equation.

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That is rufly the prosedur of solving that kind of diferential equation by diferential kalkulus. An other myns of solution is to turn the given diferential equation into a diferens equation, lIk that for the Fibonacci serys ( shown in the previus sektion ) and solv it by the kalkulus of finit diferenses:

( a,n+1 - 2a,n + a,n-1 )/h + ( 2.8(a,n - a,n-1))/h + 25a,n = 0.

Seting the intervals of tIm at h = 1/100 sekond, this bekoms:

10,000a,n+1 - 20,000a,n + 10,000a,n-1 + 280a,n - 280a,n-1 + 25a,n = 0.

This reduses to:

a,n+1 - 1.9695a,n + .972a,n-1 = 0.

In the saim way, the formula for the nth term, in the Fibonacci serys, was found. AplIing the quadratik formula to the diferens equation:

( -( -1.9695 ((-1.9695) - 4(.972)) )/2

This equals .98475 .04762i.

The imajinary term prevents an adition and subtraktion into tw ryl numbers, wich wer then rais'd to the power of n, and aded for the nth term formula for the Fibonacci serys. But the imajinary number is an operator, implying a turn thru nInty degrys, so Pythagoras' theorem yilds a hypotenus, r = .9859.

( A hypotenus, r, kan be konsider'd as the radius vektor wich swyps out a sirkl. So, it remains of konstant magnitud, wIl x and y ko-ordinats, forming the other tw sIds of a rIt-angl'd triangl, ar always chanjing. Repyted swyps of the radius kan be represented as a serys of regular wavs, in the mor familiar graf of rektilinear ko-ordinats.

Such an ordinary graf in wich the wav krests and trofs ar not uniform but dekrysing, or 'damp'd', also has a koresponding piktur, wen konsider'd as a sirkular funktion. The radius no longer swyps out the saim sirkl with ych turn, but a dekrysing sirkl, or a spiral. )

Wen r is rais'd to the power of n, it is equivalent to the exponential term, exp(mt), in the koresponding diferential equation. That is, it also represents exponential growth or, in this kais, dekay.

That is: r'n ~ ( .9859 )'n ~ ( exp(-.0142.n) ~ exp(-1.4t).

Our finit diferens equation was bais'd on a kalkulation for every 1/100 of a sekond ( h = 1/100 ). We gav an exampl of kalkulating the diferential equation solution of y, wen t = .29 or 29/100 of a sekond. In terms of the finit diferens equation, this myns syking a solution for its 29th term, or n = 29.
Yusing the diferens equation to solv for t = .29, n has to be divided by 100 ( hn = t ). To kompensat, its ko-eficient, -.0142, is multiply'd by 100, equaling -1.42, wich aproximats to, -1.4, the ryl number part of the komplex number ( and its conjugat number ) diferential equation solution for m.

Similarly for the trigonometrik terms, involving the imajinary part, of the quadratik solution, .04762, multiply'd by 100, as n is divided by 100, to bekom t.

Therfor, y = exp(-1.42t)( C kos 4.762t + D sIn 4.762t ).

Yusing the saim initial konditions, given for solving the konstants C and D in the koresponding diferential equation:

C = -.4 as befor.

And 16 = (-1.42)(-.4) + 4.762D

So, D = 3.24.

For, t = .29, the diferens equation solvs for y = 2.058 ( to thry desimal plases ). This is the saim ( to tw desimal plases ) as the mor exakt result from the diferential equation. The diferens equation's erors apyr to hav kansel'd ych other out. The korespondens betwyn the tw prosedurs isnt always quIt so klos.


Richard Lung.


Referenses:

W W Sawyer: A Path To Modern Mathematics.
This komentary is a limited suplement and by no myns a substitut for the bwk.

An introduktion to the myning of kalkulus ( wich I havnt given hyr ) kan be found in an other bwk by W W Sawyer: Mathematician's Delight ( 1943 ).

A Geary, H V Lowry, H A Hayden: Advanced Mathematics For Technical Students. Part one.




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