Convolutional Discrete Calculus – Video Outline
PART 1:
sequences, patterns, and discrete calculus
- Problem: can you find the formula for a sequence?
- Try discrete differences
- Problem: that’s the pattern, but what’s the formula?
- Write it in terms of antidifferences
- Antidifferences are indefinite sums up to a constant
- Write it in terms of sums
- Problem: that’s the sum formula, but what’s the polynomial formula?
- Summing sequences sums their differences
- Multiplying a sequence by a constant multiplies its difference by a
constant
- Use the geometry of rectangles to solve for the geometry of a
triangle
- Problem: that was hard, is there an easier way to find a more
general polynomial formula?
- List off the difference for each monomial
- Rearrange the chart in terms of sums
- Solve for each sum
- Use the chart as a reference
- Problem: can you find the level of an (easy) sequence?
- (introduce without level, then introduce level)
- Take differences until it’s constant
- Problem: can you find the level of a (medium) sequence?
- Differences don’t work
- Sums and differences cancel out
- Sums are like negative differences
- Take sums till it’s constant, and make that number negative
- Problem: finding the level of a hard sequence
- Differences don’t work, neither do sums
- The answer is
, but getting there is complicated, save it for part 2
- Problem: what exactly is a sequence’s level?
- Put simply, the level of the sequence the degree of its polynomial
expression
- Some functions, which aren’t polynomials, have no level
- Problem: what about sequences which have a degree but aren’t
polynomials?
- The polynomial definition works for polynomials, but clearly that’s
only part of a larger phenomenon
- The degree is how many sums/differences you need to apply to a
constant to get a function
- Problem: ok, but what exactly are sums and differences?
- Mess with variations on the difference definition to get a more
general “recurrence transformation” phenomonon
- Reframe the sum as a recurrence transformation
- Use weight analysis to define higher order sums/differences as
recurrence transformations
- Problem: what does this tell us about level?
- Transforming two sequences adds their level
- You can find level by splitting into
factors
- Level is just how many unit
factors the sequence can be split into
- Problem: why is the sum the unit?
- Easy answer: it lines up nicely with polynomials, but that’s not
very satisfying
- Try starting with a different base unit and base inverse
- Try finding the level of some sequences
- Everything still works, just scaled differently
- Interestingly, some sequences can now have non-integer level
- It actually is kind of arbitrary
- Problem: could we have found the sum sequence from the double sum
unit sequence?
- Easy answer: use the difference, but that’s kind of cheating
- Use a table to visualize
transforms
- Reverse the problem to solve for each item on the sides of the
table
- Vocab: functional root
- Problem: can we find the functional root of the sum sequence?
- Use the table method from before
- There’s no easy pattern, but it works!
- Problem: now can we return to the level of the hard sequence?
- Use our rules and some algebra to work out that the new sequence has
level
- Use our rules to figure out that the sequence from before has level
- Problem: start with a level and find a sequence
?
- Use the functional root technique
- Show that it works for any dyadic rational level
- Problem: can you find sequences with more numbers
()?
- Functional square roots won’t help us, since
is non-dyadic
- Functional cube roots could work for
, but calculating them is a hot mess
- Functional roots are basically hopeless for
- No clue where to even start with
- This is possible, but getting there is complicated, save it
for part 3
PART 3:
convolutions, derivatives and power series
- Problem: can we find a pattern for the weights in the
level sequence?
- Trying difference techniques won’t work
- Try on just the numerator and denominator, that still doesn’t
work
- Try and reframe the problem as shifted and scaled copies
- Notice location addition and value multiplication
- Tangent to polynomials
- Reframe polynomial multiplication as shifted and scaled copies
- Notice location addition and value multiplication
- Show equivalence between functional root problem and the polynomial
square root problem
- Demonstrate that the equivalence works by testing it on the second
order sum sequence
- Problem: ok, but how do you find a pattern?
- Use algebra to show
- Reframe the problem as power representation of
- Show that our manual calculations for the
sequence work to approximate it
- Problem: how does that help us? where’s the pattern?
- Define an analytic function, and show that
is one
- Take: calculus says that every analytic function has a power
series
- Take: the coefficients will be value, rate, rate of rate, etc
- Take: you can calculate rate by taking a derivative
- Take: you can find the derivative of any analytic function using
these rules
- Demonstrate that evaluating derivatives of
produces values of the
sequence
- Problem: ok, but how do we get from there to an actual, closed form
pattern?
- Unpack the derivative rules to show patterns in the functions
- Simplify the patterns for the special
case
- Derive a real pattern in closed form
- Review: to find a level
sequence, we took the power series of
and derived a pattern from the differentiation rules
- Problem: can we return to find sequences with weird weights?
- For
, just take the series coefficients for
- Test that it genuinely does work out to the functional cube root of
- For
, just take the series coefficients for
- You can’t test it the same way, but it does work
- Problem: what about the level i sequence?
- Complex exponentiation works with new rules
- We don’t need them though, we can just plug and chug
- Deriving a pattern
- Finding the sequence
- Demonstrating that it works
Outro
- Credits:
- Mythologer’s video
- Supware’s video
- Morphocular’s video
- Unanswered questions:
- What actually is complex exponentiation?
- What other sequences have no level?
- What if we start with a level-less unit
transformation?
- For next time:
- Can we leverage the connection between differences and derivatives
to extend our techniques to the continuous relm?
Comments
Where are the videos? On YouTube?