Domain Specific Languages of Mathematics

Transcription

1 Domain Specific Languages of Mathematics Cezar Ionescu Patrik Jansson

2 The problem Many Computer Science students seem to have difficulties with real and complex analysis. At Chalmers, the worst obstacles for students in the CSE program are two three-year courses on Signals and Systems and Control Theory. At the same time, students seem to have very little difficulty with discrete mathematics. At least in part, this seems to be due to the more CS-like style.

3 The CS style more emphasis on syntax prefer formal proofs use of logic as a tool disciplined use of ambiguity, e.g., polymorphic expressions (see Communicating Mathematics: Useful Ideas from Computer Science, Charles Wells, AMM,1995)

4 Real analysis Real analysis is usually taught in a very different style: emphasis on semantics, informal proofs, and untangling ambiguous notation can be really hard. It s easy to get it wrong: /03/brief-explanation-of-taylor-series-via.html. It s hard to get it right: taylors-theorem-with-the-lagrange-form-of-the-remainder.

5 Continuity The ɛ-δ definition of continuity: Definition X R, f : X R continuous at x X if: ɛ > 0 δ > 0 such that y X y x < δ = f y f x < ɛ

6 Continuity We can simplify the ɛ-δ definition of continuity, by introducing, e.g., neighborhoods: V : {P R} R R >0 P X V {X } x ɛ = {y y X, y x < ɛ} Definition X R, f : X R continuous at x X if ɛ > 0 δ > 0 such that {f y y V x δ} V (f x) ɛ

7 Continuity We can simplify the ɛ-δ definition of continuity, by introducing, e.g., powerset operations: P : (A B) P A P B P f X = {f x x X } Definition X R, f : X R continuous at x X if ɛ > 0 δ > 0 such that P f (V x δ) V (f x) ɛ

8 Continuity We can simplify the ɛ-δ definition of continuity, by making the dependency of δ on ɛ explicit: Definition X R, f : X R continuous at x X if δ : R >0 R >0 such that ɛ > 0 P f (V x (δ ɛ)) V (f x) ɛ

9 Continuity We can simplify the ɛ-δ definition of continuity, by making the dependency of δ on ɛ explicit and using function composition: Definition X R, f : X R continuous at x X if δ : R >0 R >0 such that ɛ > 0 (P f V x δ) ɛ V (f x) ɛ

10 Continuity We can simplify the ɛ-δ definition of continuity, by making the dependency of δ on ɛ explicit, using function composition, and lifting to functions: Definition X R, f : X R continuous at x X if δ : R >0 R >0 such that: P f V x δ V (f x)

11 Properties of V and P Properties of V : V x increasing X Y = V {X } x V {Y } x Working with neighborhoods instead of explicit the ɛ δ machinery ensures a degree of implementation abstraction. Properties of P: P is a (co-variant!) functor P f increasing

12 Calculational proofs If f : R R continuous at x and g : R R continuous at f x, then g f continuous at x. We are given and P f V x δ f V (f x) P g V (f x) δ g V (g (f x)) and we have to find δ g f : R >0 R >0 such that P (g f ) V x δ g f V ((g f ) x)

13 Calculational proofs g continuous at f x {definition} P g V (f x) δ g V (g (f x)) {f continuous at x, monotonicity} P g (P f V x δ f ) δ g V (g (f x)) {P functor} P (g f ) V x (δ f δ g ) V ((g f ) x)

14 Limit points Definition x limit point of X if every neighborhood of x contains a point x x such that x X.

15 Limit points Definition x limit point of X if every neighborhood of x contains a point x x such that x X. Introducing neighborhoods: x limit point of X if ɛ > 0 x V {X } ɛ such that x x.

16 Limit points Definition x limit point of X if every neighborhood of x contains a point x x such that x X. Introducing neighborhoods: x limit point of X if ɛ > 0 x V {X } ɛ such that x x. Using the implicit parameter of V instead of the : x limit point of X if ɛ > 0 x V {X {x}} ɛ.

17 Limit points Definition x limit point of X if every neighborhood of x contains a point x x such that x X. Introducing neighborhoods: x limit point of X if ɛ > 0 x V {X } ɛ such that x x. Using the implicit parameter of V instead of the : x limit point of X if ɛ > 0 x V {X {x}} ɛ. Using instead of : x limit point of X if ɛ > 0 V {X {x}} ɛ.

18 Limit points Definition x limit point of X if every neighborhood of x contains a point x x such that x X. Introducing neighborhoods: x limit point of X if ɛ > 0 x V {X } ɛ such that x x. Using the implicit parameter of V instead of the : x limit point of X if ɛ > 0 x V {X {x}} ɛ. Using instead of : x limit point of X if ɛ > 0 V {X {x}} ɛ. Lifting point-wise to functions: x limit point of X if V {X {x}}.

19 Limits of functions Definition Let f : X R and p a limit point of X. We write f x q as x p, or lim (f x) = q x p if q R such that ɛ > 0, δ > 0 such that f x q < ɛ for all x X such that 0 < x p < δ.

20 Limits of functions Definition Let f : X R and p a limit point of X. We write f x q as x p, or lim (f x) = q x p if q R such that ɛ > 0, δ > 0 such that for all x V {X {p}} p δ f x V q ɛ

21 Limits of functions Definition Let f : X R and p a limit point of X. We write f x q as x p, or lim (f x) = q x p if q R such that ɛ > 0, δ > 0 such that P f (V {X {p}} p δ) V q ɛ

22 Limits of functions Definition Let f : X R and p a limit point of X. We write f x q as x p, or lim (f x) = q x p if q R and δ : R >0 R >0 such that ɛ > 0: P f (V {X {p}} p (δ ɛ)) V q ɛ

23 Limits of functions Definition Let f : X R and p a limit point of X. We write f x q as x p, or lim (f x) = q x p if q R and δ : R >0 R >0 such that: P f V {X {p}} p δ V q

24 Limits of functions Definition Let f : X R and p a limit point of X. lim{x }f p = q if q R and δ : R >0 R >0 such that: P f V {X {p}} p δ V q

25 Limits of functions An important property: Let f : X R, lim f p = q, Y X such that p is a limit point of Y, then lim {Y } f p = lim {X } f p = q We have: P f V {X {p}} p δ V q. We calculate: Y X = { increasing} Y {p} X {p} = {V increasing in implicit argument} V {Y {p}} p V {X {p}} p = {P f increasing} P f V {Y {p}} p δ P f V {X {p}} p δ = {transitivity of } P f V {Y {p}} p δ V q

26 A calculus of limits If lim f x and lim g x are defined, then lim (f + g) x = lim f x + lim g x lim (f g) x = lim f x lim g x if B 0, then lim f g x = lim f x lim g x if f 0, then lim f x 0

27 Limits of functions Application: if x is a limit point of X, then f : X R continuous at x iff lim f x = f x. f continuous at x {definition} P f V {X } x δ V (f x) {monotonicity of V and P f, f x V (f x)} P f V {X {x}} x δ V (f x) {definition} lim f x = f x In particular, this ensures that we also have a calculus of continuous functions (they can be added, multiplied, etc.).

28 Differentiability Definition Let f : [a, b] R. For any x [a, b] form the quotient and define φ t = f t f x t x (a < t < b, t x) f x = lim φ x provided this limit exists. The (partial) function f is called the derivative of f. If f is defined at a point x, we say that f is differentiable at x.

29 Differentiability We can shorten the definition by using the overloading of operations for functions: Definition Let f : [a, b] R. For any x [a, b] define f x = lim {[a, b] {x}} f f x id x provided this limit exists. The (partial) function f is called the derivative of f. If f is defined at a point x, we say that f is differentiable at x. x

30 Differentiability and continuity If f : [a, b] R is differentiable at x [a, b], then f is continuous at x. Proof (Rudin) As t x, we have: f t f x = f t f x t x (t x) f x 0 = 0

31 Differentiability and continuity f continuous at x {Every x [a, b] is a limit point of [a, b]} lim {[a, b]} f x = f x {x is a limit point of [a, b] {x} [a, b]} lim {[a, b] {x}} f x = f x {Calculus of limits: lim k x = k} lim f x = lim (f x) x {arithmetic, calculus of limits: lim f x lim g x = lim (f g) x} lim (f f x) x = 0

32 Differentiability and continuity lim (f f x) x = 0 {Arithmetic in [a, b] {x}} lim ( f f x (id x)) x = 0 id x {Calculus of limits: lim f x lim g x = lim (f g) x} lim f f x x lim (id x) x = 0 id x {Definition of f x, calculus of limits: lim id x = x} f x 0 = 0 {Arithmetic} true

33 Differentiability and optimality Definition Let f : X R. We say that f has a local maximum at x X if there exists δ > 0 such that f x f x for all x with x x < δ. Theorem: Let f : [a, b] R; if f has a local maximum at x [a, b], and if f x exists, then f x = 0.

34 Differentiability and optimality Definition Let f : X R. We say that f has a local maximum at x X if there exists δ > 0 such that f x f x for all x V x δ.

35 Differentiability and optimality Definition Let f : X R. We say that f has a local maximum at x X if there exists δ > 0 such that f x f x for all x V x δ. Theorem: Let f : [a, b] R; if f has a local maximum at x (a, b), and if f x exists, then f x = 0.

36 Differentiability and optimality lim f f x x id x ={Making the implicit parameter explicit} lim {[a, b] {x}} f f x x id x ={x is limit point of [a, x) [a, b] {x}} lim {[a, x)} f f x x id x {Calculus of limits: f f x 0 on [a, x)} id x 0

37 A calculus of continuous and differentiable functions We have mentioned that a calculus of continuous functions extends that of limits. Additionally, we have for every continuous f : [a, b] R: m, M R such that P f [a, b] = [m, M] If f, g : [a, b] R differentiable at x [a, b], then f + g, f g, f /g are differentiable at x (the last assuming g x 0, and (f + g) x = f x + g x (f g) x = f x g x + f x g x ( f g ) x = f x g x f x g x g 2 x Derivatives of common functions: k = 0, (x n ) = n x n 1, etc.

38 Rolle s theorem Let f : [a, b] R, continuous on [a, b] and differentiable on (a, b), such that f a = f b = 0. Then there exists x (a, b) such that f x = 0. Proof: f continuous on [a, b] = {Calculus of continuous functions} P f [a, b] = [m, M] = {f a P f [a, b]} m 0 M If m = M then f = 0, therefore f x = 0 for any x (a, b).

39 Rolle s theorem If m < M then m 0 M 0. Consider M 0: M 0 = {f a M f b M M P f [a, b]} x (a, b) such that f x = M = { x [a, b] : f x M} x maximum = {Derivative at the maximum} f x = 0

40 The mean value theorem: the problem Let f : [a, b] R, continuous on [a, b] and differentiable on (a, b).

41 The mean value theorem: the problem Let f : [a, b] R, continuous on [a, b] and differentiable on (a, b). Assume that we are given f a and a way of computing f x, but not other values of f, and we want to estimate f b. This is actually often the case in numerical computations, e.g. in interval analysis.

42 The mean value theorem: the problem Let f : [a, b] R, continuous on [a, b] and differentiable on (a, b). Assume that we are given f a and a way of computing f x, but not other values of f, and we want to estimate f b. This is actually often the case in numerical computations, e.g. in interval analysis. Basically, we are looking for some x and some α, such that f x = α can tell us something about the relationship between f b and f a.

43 The mean value theorem: the problem Let f : [a, b] R, continuous on [a, b] and differentiable on (a, b). Assume that we are given f a and a way of computing f x, but not other values of f, and we want to estimate f b. This is actually often the case in numerical computations, e.g. in interval analysis. Basically, we are looking for some x and some α, such that f x = α can tell us something about the relationship between f b and f a. This might seem impossibly vague, but we can compute:

44 The mean value theorem: the computation x : f x = α = {The only qualitative result we know} x : g x = 0 = f x α = {Rolle s theorem, differential calculus} g a = g b = 0 g x = f x α x β = {Arithmetic} α = f b f a b a β = f a b f b a b a

45 The mean value theorem We have proven the mean value theorem: (Rudin) Let f : [a, b] R, continuous on [a, b] and differentiable on (a, b). Then there exists x (a, b) such that f x = f b f a b a. Equivalently: f b = f a + f x (b a) If M is an upper bound for f x, then M (b a) is an upper bound for f b f a.

46 Taylor s theorem: the idea Can we do better if f is twice differentiable? We can carry out a similar computation: x : f x = α = {The only qualitative result we know} x : g x = 0 = f x α = {Rolle s theorem, differential calculus} x : g a = g x = 0 g x = f x α x β = {Rolle s theorem, differential calculus} g a = g b = 0 g x = f x α 2 x 2 β x γ = {Arithmetic} f b f a f a (b a) = α β =..., γ =... (b a)2 2

47 Taylor s theorem: the idea We have proven that x (a, b) such that f b = f a + f a (b a) + f x (b a)2 2 In general, this is a better approximation of f b. We can now easily generalize the result to Taylor s theorem (with Lagrange rest) and prove it by induction (exercise!).

48 Future work Implementation. Extending to Fourier series and Laplace transforms; some linear algebra (e.g., for the conjugate gradient method). Investigate related projects, e.g.: Abstract Math (Wells), Functional Differential Geometry (Sussman, Wisdom, Farr), Funmath (Boute et. al.).