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Let D be an open subset of |R^2.

Let f : D -> |R.

Suppose that the partial derivatives of f exist on D.

Let (a,b) be a point in D.

Suppose that fx(a,b)=fy(a,b)=0.

Does it mean that f has an extremum at (a,b)?

Let f : D -> |R.

Suppose that the partial derivatives of f exist on D.

Let (a,b) be a point in D.

Suppose that fx(a,b)=fy(a,b)=0.

Does it mean that f has an extremum at (a,b)?

NO.

f(x,y)=x^2-y^2 at (0,0)

f(x,y)=x^2-y^2 at (0,0)

Let X be a set, and W c P(X).

Decide and prove which is true:

W* is the _________ collection of sets that generates W- convergence.

(1) largest

(2) smallest

Decide and prove which is true:

W* is the _________ collection of sets that generates W- convergence.

(1) largest

(2) smallest

W* is the largest collection of sets that generates W-
convergence.

page 51 in gen top

page 51 in gen top

Let X be a set, and W c P(X).

Prove that (W*)* c W*.

Prove that (W*)* c W*.

Use the fact that W-convergence implies W*-convergence.

page 52 in gen top

page 52 in gen top

X, W c P(X)

What does it mean that W is a semi-ring?

What does it mean that W is a semi-ring?

(1) W is non-empty

(2) A,B:-W => AnB:-W

(3) A,B:-W => A\B is a finite disjoint union of sets from W

page 60 in 1st measure

(2) A,B:-W => AnB:-W

(3) A,B:-W => A\B is a finite disjoint union of sets from W

page 60 in 1st measure

Let W c P(X) be a semi-ring.

Does W have to contain the empty set?

Does W have to contain the empty set?

Yes.

W is non-empty, hence A:-W.

Then A\A is a finite union of sets from W.

Hence W has to contain the empty set.

page 60 in 1st measure

W is non-empty, hence A:-W.

Then A\A is a finite union of sets from W.

Hence W has to contain the empty set.

page 60 in 1st measure

X, J c P(X)

J is a semi-ring.

Prove that Js = R(J).

Js denotes the collection of all finite unions of sets from J.

R(J) denotes the ring generated by J.

J is a semi-ring.

Prove that Js = R(J).

Js denotes the collection of all finite unions of sets from J.

R(J) denotes the ring generated by J.

Show that Js is a ring.

HINTS:

A\B :- Jsds

Jd = J

Jsds = Jdss = Jds = Js

HINTS:

A\B :- Jsds

Jd = J

Jsds = Jdss = Jds = Js

Let W c P(X) satisfy:

(1) /\B,A:-W B u A :- W

(2) /\B,A:-W B n A :- W

Does W have to be a ring?

(1) /\B,A:-W B u A :- W

(2) /\B,A:-W B n A :- W

Does W have to be a ring?

NO.

W = { {1} }

W = { {4}, {4,5} }

W = { O, {6}, {6,7} }

W = { {1} }

W = { {4}, {4,5} }

W = { O, {6}, {6,7} }

sup { sqrt(a/b) / (a + 1/b) : a,b>0 } = ???

= 1/2

Let 0 < b < a.

log(a/b) < ???

log(a/b) < ???

0 < b < a ==> log(a/b) < (a-b)/b

page 153 in golden gate

page 153 in golden gate

Let 0 < b < a.

log(a/b) > ???

log(a/b) > ???

0 < b < a ==> log(a/b) > (a-b)/a

page 153 in golden gate

page 153 in golden gate