Advice for Research Mathematics

We discuss the role of composition and compositional inverses in the world of polyseries. Composition is an important kind of additional operation that is available for polynumbers, and the notion of a compositional inverse becomes available once we move further into polyseries.

There is an important argument here that shows how to inductively determine such inverses when they exist.

Here are the Wild Egg Maths Playlists (some available only to Members!)

Algebraic Calculus One (for the Open Learning Course) : https://www.youtube.com/playlist?list...

Algebraic Calculus Two (for the Open Learning Course in preparation): https://www.youtube.com/playlist?list...

Advice for Mathematics Exploration (for Members only): https://www.youtube.com/playlist?list...

Solving Polynomial Equations (Members only): https://www.youtube.com/playlist?list...

Hexagrammum Mysticum (Members only): https://www.youtube.com/playlist?list...

Algebraic Calculus and de Casteljau Bezier curves (Members only): https://www.youtube.com/playlist?list...

Exceptional Structures in Maths and Physics via Dynamics on Graphs (Members only): https://www.youtube.com/playlist?list...

Maxel Inverses and Orthogonal Polynomials (for Members only): https://www.youtube.com/playlist?list...

Triangle Centres: https://www.youtube.com/watch?v=iLBGX...

Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list...

*

Here are the Insights into Mathematics Playlists:

Maths and Music: https://www.youtube.com/playlist?list...

Diffusion Symmetry: https://www.youtube.com/playlist?list...

Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist?

list=PL8403C2F0C89B1333

Playing Go: https://www.youtube.com/playlist?list...

Year 9 Maths: https://www.youtube.com/playlist?list...

Ancient Mathematics: https://www.youtube.com/playlist?list...

Wild West Banking: https://www.youtube.com/playlist?list...

Sociology and Pure Mathematics: https://www.youtube.com/playlist?list...

Old Babylonian Mathematics (with Daniel Mansfield): https://www.youtube.com/playlist?

list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov

Math History: https://www.youtube.com/playlist?list...

Wild Trig: Intro to Rational Trigonometry: https://www.youtube.com/playlist?list...

MathFoundations: https://www.youtube.com/playlist?list...

Wild Linear Algebra: https://www.youtube.com/playlist?list...

Famous Math Problems: https://www.youtube.com/playlist?list...

Probability and Statistics: An Introduction: https://www.youtube.com/playlist?list...

Boole's Logic and Circuit Analysis: https://www.youtube.com/playlist?list...

Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list...

Differential Geometry: https://www.youtube.com/playlist?list...

Algebraic Topology: https://www.youtube.com/playlist?list...

Math Seminars: https://www.youtube.com/playlist?list...

0:01

Hello everyone I'm Norman Wildberger thanks for joining me on this adventure

0:06

exploring research level mathematics I'm giving you some advice I'm trying to expand your thinking

0:12

and today we want to enrich our augmented number Theory point of view so we've gone from arithmetic

0:19

with ordinary numbers to arithmetic with poly numbers and then poly series and then

0:25

another big step to Max Souls so this is a kind of an enlargement of number Theory but

0:30

it ends up touching with important aspects from special functions from the theory of orthogonal

0:38

polynomials linear algebra certain interesting matrices that arise and also today calculus as

0:45

we touch base with the exponential and log series okay in a particular form so I want to introduce

0:54

these um to you in the context of what we might call compositional inverses for poly Series so

1:02

it turns out that the world of poly series is in some important ways much richer than the

1:09

arithmetical world of ordinary numbers and one reason for that is that the ordinary operations

1:17

of addition multiplication division etc that hold for numbers can be extended to poly numbers

1:25

and poly series but in addition there's an extra new operation at that level which is not visible

1:34

on the ordinary number Theory arithmetic and that is composition So today we're going to

1:41

talk about the implications of introducing this new operation to augment the landscape very

1:48

considerably okay this is a hugely important uh sort of new aspect to this this world

1:55

okay so I'm going to also make some shifts a little bit in notation I'm moving us so just

2:02

gently towards the algebraic calculus kind of notation so in particular in the world of

2:08

poly numbers we like to think of poly numbers as vertical lists of numbers okay and in that

2:14

world this particular poly number whose name is Alpha maybe I should put a definition here

2:21

um there this poly number with entries zero one that's a distinguished poly number which is uh

2:29

sort of at the heart of polynomial Theory it's sort of the simplest poly number which is not a

2:34

regular number so if there's just a single entry in the top spot then we're essentially talking

2:39

about an ordinary number so this is sort of the simplest poly number which is not an ordinary

2:45

number okay so we're going to use this Alpha and um and then I'm going to Define a property that

2:54

um a general poly number or in fact a poly Series has we'll say it's grounded precisely when the

3:01

top entry is zero like this one here so this is a grounded poly number because the top entry is zero

3:09

so the grounded poly numbers or grounded poly series uh play a special role okay and one of

3:16

the reasons is because of the following statements that if p is a poly series

3:21

that means it's a poly number that keeps on going in some fashion or another

3:27

then we can evaluate it on a grounded poly number Q in fact we can go further and we can evaluate

3:36

it even on a grounded poly series so P of Q This composition is still defined so I want to

3:47

illustrate that and let's illustrate it with one of the most important poly series The exponential

3:52

poly series okay so X where X Alpha is this poly series here summation one over K factorial Alpha

4:01

to the k k going from zero onwards okay and um and let's also just gently introduce the small

4:13

distinction between lower dots like these dots here and Central dots like these ones here so

4:21

in the algebraic calculus series it becomes important to look more carefully at series than

4:28

than we usually do and I think a big distinction to be made there is the distinction between

4:35

knowing that we have a series with a pattern whose whose formula or whose algorithm is known to us

4:44

and having a series which goes on in some fashion perhaps but we don't know how it actually proceeds

4:52

so I would like to sort of gently make this distinction by reserving lower dots

4:58

generally to represent uh an odd series whose pattern or formula or algorithm is known to us

5:05

and the center dots formulation to represent on series whose terms Beyond a certain point are

5:12

unknown to us okay so in this case here we do have a formula here's the formula so it's appropriate

5:19

to use lower dots indicating that we know how this thing goes on okay okay so let's take

5:26

this on series and let's take this poly series just ordinary Q just Alpha plus 3 Alpha squared

5:34

so as a poly series that would be zero one three in particular that's a grounded

5:42

poly number because the top entry is zero okay so now what happens if we try to

5:49

plug in Q for for Alpha in other words evaluate P this exponential poly series at Q

5:58

so we take this series here and we replace all the Alphas with cubes so we get 1 plus Alpha Plus

6:06

or Alpha plus 3 Alpha squared okay that's the result of replacing this Alpha with a Q and then

6:11

we get one half here's Q squared plus 1 6 Q cubed plus 1 24th Q to the fourth etc okay now when we X

6:22

these individual terms The crucial thing to observe here is that say any one of them say let's

6:27

say we expand this thing out here right since what's inside here has no constant term the lowest

6:33

term is an alpha that means that when we expand this to the fourth power the lowest power of alpha

6:41

that we're going to get is Alpha to the four we're not going to get any terms less than Alpha to the

6:46

4 in degree here we're just going to get Alpha to the fifth terms and higher etc now the effect of

6:55

that is that as we look through here and say we're interested in uh you know finding all the alpha

7:01

cubed terms okay the alpha Cube terms well there's going to be um you know possible contributions uh

7:08

from here all the way up to here up to the third power but no further there are no Alpha cubed

7:17

contributions from all the terms from here on further in particular that means that the

7:23

total Alpha cubed contribution is finite and we can calculate that coefficient in this case it

7:31

turns out to be 19 over 6 okay so there's some contribution coming from this term there's also

7:36

a contribution coming from this term and that's it okay so the resulting series is one plus Alpha

7:42

plus seven over two alpha squared plus 19 over 6 Alpha cubed etc and I'm putting lower Dots here

7:48

because although we don't actually have a formula an explicit formula for this exactly we do have a

7:55

pretty clear algorithm right we could we could in principle sort of calculate any one of these

8:00

things because we know that it's coming from this this uh this series here with the substitution

8:06

foreign let's also observe that if we did something different if we thought about

8:13

replacing Q with just a number like eight okay what's P of eight well that is not making sense

8:23

for us because then we have a whole bunch of numbers and we're being asked to add up all

8:29

these ongoing a number of numbers and as opposed to my the rest of the mathematical Community okay

8:36

I happen to not believe that you can add up an an infinite number of numbers whether these things

8:43

are so-called converging or not okay that's beside the point the issue is you're not able

8:49

to perform an infinite number of arithmetical operations so it's a meaningless quantity okay

8:55

so P of 8 is as far as I'm concerned meaningless except in an approximate setting right an applied

9:03

mathematician could say okay what do we really mean by that is we're just going to go up to up

9:07

to these terms here and truncate the rest well fine then it becomes a finite doable calculation

9:13

but in the context of pure maths that doesn't make sense but P of Q does make sense if Q is

9:22

grounded okay so it's exactly the existence of having a number here in that top spot which

9:30

determines whether you're able to evaluate a general poly series at such a thing or not

9:37

okay so having gotten that out of the way we can now Define the idea of a compositional inverse

9:45

so we've been talking about inverses a lot so we have inverses of

9:49

poly series you know if you have a poly series you might look at one over that poly series and

9:56

and again there's a condition um in that case actually you require a constant term okay so

10:02

if you have a constant term then the inverse of a poly series makes sense as is another poly series

10:09

another important place where inverses take place is with the in the context of matrices

10:14

right we've seen the important role of taking inverses of matrices or maxims here now we're

10:21

going to introduce another notion of inverse the idea of a compositional inverse because we have

10:27

this new operation the operation of composition and so we can inquire if we have a poly series p

10:35

when can we find another poly series Q so that P of Q equals Alpha essentially the identity

10:46

poly series and in fact we require P of Q to be Alpha and also in the other order

10:52

Q of P equals Alpha so if we can find a q another poly series for which both of

11:00

these Expressions evaluate to Alpha then we say that P has a compositional inverse of Q

11:09

then it's symmetrical then the compositional inverse of Q will be p

11:15

okay so now there's a sort of an important theorem here the theorem is that a poly series p

11:21

say a0 plus A1 Alpha plus A2 Alpha squared etc has a compositional inverse Q which is also a

11:30

poly series precisely when a0 equals zero in other words when it's grounded and when A1

11:43

the coefficient of alpha is non-zero so that's the Criterion that will guarantee that we have

11:53

an inverse and in such a case the inverse Q is unique and the terms may be calculated inductively

12:00

foreign so this is a very important kind of Direction and in the algebraic calculus 2

12:08

course we're going to be looking at this concept in in considerable detail and there's there's

12:13

lots of really rich examples here so we're only going to be just touching a base with

12:18

this very important topic so let me illustrate this okay so here is um a poly series P equals

12:25

Alpha plus 2 Alpha squared minus Alpha cubed plus 3 Alpha to the fourth plus

12:30

and then it goes on in some fashion you see these Central Dots here these Central dots are my way

12:37

of indicating that how this thing goes on is not clear to us okay we're not given a pattern we're

12:45

not given a formula there's no program that we can look up that will tell us how we're

12:50

supposed to extend this we're just saying that this thing goes on possibly in some fashion

12:55

and how far it carries on you know and what what the nature of these terms are we just don't know

13:02

but we can still ask about a compositional inverse to this thing up to the level that

13:09

we can see things namely up to degree four so a compositional inverse in this context will be

13:16

another poly series say up to degrees four with say coefficients B1 Alpha plus B2 Alpha squared

13:23

plus B3 Alpha Cube plus B4 Alpha to the fourth notice this thing here is grounded right doesn't

13:28

have any constant term and I'm positing that the inverse also doesn't have a constant term

13:35

and what we're trying to arrange is that if we take p and we evaluate it at Q in other words

13:41

we replace the alpha here with q and expand that at least up to terms of degree four we're going

13:50

to get Alpha so let's see what's involved here and that will I think give you a pretty clear idea at

13:57

least uh you know a direction of thinking about why this theorem is is likely uh reasonable so

14:03

let's look at this equation here and uh and sort of substitute things in so here's P we're going to

14:08

replace Alpha with this Q okay so there's um the the right hand side I'm bringing it over to the

14:15

left hand side so we have Alpha equals okay here's Q plus two here's Q squared minus Q cubed plus 3

14:27

Q to the fourth and then there's other additional terms but they're all of degree bigger than four

14:35

all right so if we take this thing here and we expand the right hand side up to terms

14:43

of degree four okay so when we when we expand this there will be of course terms which are of

14:50

higher degree we just ignore them so we're only tabulating things up to Alpha to the four and

14:56

disregarding everything else okay so what do we get uh well we get this thing here um yes we're

15:03

just taking this thing here and evaluating it and then you get a poly series an alpha and the lowest

15:10

degree terms Alpha to the one because there's no constant term floating around here anywhere

15:16

so we get an alpha times B1 so this is the only linear term Alpha squared 2b1 squared plus B2

15:26

so coming from from this thing here squared and also from this thing here and then plus Alpha

15:32

cubed and here's the coefficient of alpha cubed okay obtained from something here or something

15:37

here something here plus Alpha Four here's the coefficient of alpha the four okay and all of

15:42

this should be equal to Alpha that's a poly number in Alpha that's a poly number in Alpha

15:49

so the coefficients have to agree so then we have to solve so the B1 had better be equal to one for

15:56

things to agree this coefficient here better be equal to zero this coefficient better be equal

16:00

to zero and this coefficient better be equal to zero now if we look at these four equations we

16:07

can see that we can successively evaluate the B's starting with the first one that gives us

16:14

B1 once we know B1 then the second equation gives us B2 because the B2 is appearing only linearly

16:21

then once we have B1 and B2 the third equation which is non-linear in B1 B2 but it's linear in B3

16:30

so once we have B1 and V2 we can calculate B3 and then finally the fourth one we can calculate B4

16:39

and there's a unique solution okay and the unique solution happens to be B1 is one

16:45

B2 is -2 B3 is 9 and B4 is minus 53. so putting that together this compositional inverse of p is

16:56

Q is Alpha minus 2 Alpha squared plus 9 Alpha cubed minus 53 Alpha to the fourth and

17:06

and then more okay but we we don't know anything about the additional terms because we don't know

17:11

anything more about the additional terms of P so that's as much we can say about Q let's

17:21

just check this all right so let's take this p and substitute these terms here okay and then evaluate

17:29

and we do find that we get Alpha plus terms of degree 5 or higher and we want to also check the

17:37

other way around if we take Q this one here and now substitute the values of P right so that's

17:44

there's P minus 2 p squared plus 9 P cubed minus 53 P to the fourth we evaluate that we're going

17:53

to get Alpha plus terms of degree 5 or higher so p and Q are indeed compositional inverses

18:07

are pretty non-trivial kind of calculation you know to go from this thing here Alpha to Alpha

18:14

squared minus Alpha cubed plus three Alpha to the fourth two the inverse being Alpha minus two alpha

18:20

squared plus nine Alpha cubed minus 53 Alpha the fourth you know it'd be hard to Guess that

18:25

that's something that you have to calculate in this inductive step-by-step procedure and

18:31

you know the answer is is is not clear before you actually do it

18:36

so this has the same kind of richness that's taking an inverse of a matrix right we're getting

18:43

something non-trivial you're getting something that uh that will typically surprise us perhaps

18:50

there's informational content in this shift from uh grounded poly series to its compositional

18:58

inverse not to be confused with its reciprocal okay so if we have a poly series now you know

19:05

maybe we can ask what doesn't have a reciprocal and what's what's the formula or the the form of

19:10

the reciprocal or we can ask what's the form of its compositional inverse and admittedly

19:16

it's a bit confusing because the term inverse uh somehow sort of sometimes straddles both of these

19:22

um you know interpretations okay so now we have this really new powerful tool of compositional

19:30

inverses and in our next video I'm going to show you how this naturally brings about the

19:38

the logarithm poly series or a closely related relative in the context of the the world of

19:46

compositional inverses so we're going to see that this sort of abstract uh

19:51

you know algebra with poly series actually has a direct application to the world of calculus.