M.E.T.U.
DEPARTMENT OF
METALLURGICAL &
MATERIALS
ENGINEERING
METE 206
PROPERTY CONTROL BY HEAT TREATMENT
(EXPERIMENT 2)
STUDENT NAME
: ALI ATE
STUDENT NUMBER : 1393933
GROUP NUMBER
:2
OBJECTIVE:
To apply different cooling rates to various type of steels a
topological isomorphism, also called a homeomorphism. Ley U and V be as above and let W be
an open subset of a Banach space G. If f : U V and g : V W are C k isomorphisms,
then g f : U W is a C k isomorphism. A linear C 0 isomorphism u L(E, F) is a C i
can show (Theorem 3.5) that M2 t [M]t is a martingale. Now, when H is simple, it is not hard
to convince yourself that the integral (2.1) must also be a martingale. So what should be the
quadratic variation of R t 0 HsdMs ? Based on the approximation (2.3
2 i . find the extrema of f on the unit sphere S n1 = cfw_x R n ; Pn i=1 x 2 i = 1. Exercise 1.5.
Find the local extrema of the following functions: (i) f(x, y) = x 2 xy + y 2 2x + y, (ii) f(x, y)
= x 2 y 3 (6 x y), (iii) f(x, y) = x 3 + y 3 3xy, (iv) f(x
Inductively, given (X n1 d , d Dn1, we build (Xn d , d Dn) in such a way that (Xn d , d
Dn) satisfies (i), (ii), (iii) in the definition of the Brownian motion (where the instants under
consideration are taken in Dn). To this end, take d Dn \ Dn1, and le
(fn(x)n has a limit f(x), (ii) the sequence (fn)n converges to f uniformly on any ball of E
contained in U, (iii) the function f : x 7 f(x) is differentiable and f = g. Proof. (A) First,
assume that U is an open ball of radius R and denote by B this ball.
(1)(a) = f (a) L(E, F). Hence, f (n1) is a map V L n1 (E, F) and f (n) (a) L n (E, F)
by (2.1). One sets for short f = f (2) . For x, y E we write f (a)xy instead of f (a)(x, y).
Theorem 2.1.3. (The Schwarz Lemma.) If f (n) (a) exists, then the multilinea
Stochastic Calculus and Applications Nathanael Berestycki Cambridge These notes and other
information about the course are available on
www.statslab.cam.ac.uk/beresty/teach/StoCal/stocal.html Christina Goldschmidt, Stefan
Grosskinsky, Gregory Miermont, an
hence g is constant q.e.d. Corollary 1.3.5. Let f : U F be a map and assume that f is
differentiable on U. Let a U and let > 0 such that the closed ball B(a, ) is contained in U.
Then for any h E with h , one has f(a + h) f(a) h sup 0t1 f (a + th)
as for the circle that T 2 is a submanifold of dimension 2. (iii) A graph f U F, as in Example
3.3.5 (ii), may also be obtained by the immersion E U
U F, x 7 (x, f(x). 3.4 Tangent
space Definition 3.4.1. Let E be a finite dimensional real vector space a
BTa = a for every a. We thus have: P(St a, Bt b) = P(Ta t, Bt b) = P(Ta t, B(Ta) tTa b
a) = P(Ta t, B (Ta) a b). Now, by the strong Markov property at time Ta, BTa is a
Brownian motion independent of FTa and thus of Ta. In particular, we deduce that the
2p K + 1 n o Y K j=2 P N (0, 1/n) 2p K + 1 n P N (0, 1) 2p K + 1 n n 1/2
K1 2p K + 1 n n 1/2 K1 = C n(1/2)(K1) It follows that for all n (K + 1)/ : mnX1
i=0 P(Ai,n) Cm n(1/2)(K1)1 Thus if K is large enough that ( 1/2)(K 1) > 1, the righthand side ten
example, consider E = R 2 , a = (0, 0), S = cfw_(x, y) R 2 ; y x 2 = 0 and f(x, y) = x 2 y.
Then TaS = cfw_(x, y); y = 0, f (a)TaS = 0, f (a)TaS(x, x) = x 2 is positive definite, but f has not
a strict minimum at a on S. Exercises to Chapter 3 Exercise
D(x1,.,xp) D(f1,.,fp) D(xp+1,.,xn) 0 Inp ! . Therefore, is a C k isomorphism of R n in a
neighborhood W of a by Theorem 3.1.3. Set G = R np and denote by : R p R np R p
the projection, we have f = . q.e.d. Theorem 3.2.5. Let E U f F be a map of class C k
Otherwise said, (B (n) t1 , . . . , B(n) tk ) converges in distribution to (G1, G2, ., Gk), which is a
random vector whose law is characterized by the fact that (G1, G2G1, ., GkGk1) are
independent centered Gaussian random variables with respective covari
n! (n+1)(t)dt. Since (p) (t) = f (p) (a + th) h p , we get the result. q.e.d. 2.2. THE TAYLOR
FORMULA 37 Let us consider the particular case where E = R n . In this case f P (a) h = n i=1
hi f xi (a). Denote by h the differential operator h = Xn i=1 hi xi
Assume that u (a) L(E, L(F1, F2) and v (a) L(E, L(F2, F3) exist. Then w (a) L(E,
L(F1, F3) exists and for h E, one has: w (a) h = (v (a) h) u(a) + v(a) (u (a) h). Proof.
The proof is left as an exercise. q.e.d. Proposition 2.1.9. Let E, F, G be normed spa
X nk+1 t () X nk t () < . 1Usual conventions about versions apply, dealing with equivalence classes analogous to L p spaces 3 THE STOCHASTIC
INTEGRAL 32 Since the space of cadlag functions equipped with the k k norm is complete, there exists a c`adl`agpr
x (x, g(x) for (x, y) V . 48 CHAPTER 3. SUBMANIFOLDS Proof. Consider the map fe: U
E G, (x, y) 7 (x, f(x, y). The differential is given by the matrix fe (x, y) = idE 0 f x f
y . This matrix is invertible at (a, b). Indeed, recall that for an invertible
wanted properties. Finally, it is straightforward to build a Brownian motion in Rd, by taking d
independent copies B1 , . . . , Bd of B and checking that (B1 t , . . . , , Bd t ), t 0) is a Brownian
motion in R d . Remark. The extension of (Bd, d D) could
PROPERTY CONTROL BY HEAT TREATMENT
Objective:
To apply different cooling rates to various type of steels and to see the effect of cooling rate
and steel composition on final hardness values.
Apparatus:
 A muffle furnace.
 A hardness indenter.
 A water

Normalized (air cooled) specimen has less proeutectoid ferrite in it since cooling rate is
faster and there is not enough time to diffusion. Therefore, there is more pearlite and it is finer
than annealed specimen. As a result, normalized specimen is ha
M.E.T.U.
DEPARTMENT OF
METALLURGICAL &
MATERIALS
ENGINEERING
METE 206
PROPERTY CONTROL BY HEAT TREATMENT
(EXP 2)
Erdem MERMER
1447929
GROUP 10
OBJECTIVE:
To apply different cooling rates to various type of steels and determine the effect of cooling
rates
Assistant : Kemal Davut
METE 206 MATERIALS LABORATORY
PROPERTY CONTROL BY HEAT EXPERIMENT
Hakan Aktulga
1393875
Group 2
PROPERTY CONTROL BY HEAT TREATMENT
Objective:
To apply different cooling rates to various type of steels and to see the effect of cool
M.E.T.U.
DEPARTMENT OF
METALLURGICAL &
MATERIALS
ENGINEERING
METE 206
PROPERTY CONTROL BY HEAT TREATMENT
(EXP 2)
Erdem MERMER
1447929
GROUP 10
OBJECTIVE:
To apply different cooling rates to various type of steels and determine the effect of cooling
rates
a(k + 1)2n ) a(k2 n ) . (2.8) Then v(t) := limn v n (t) exists for all t 0 and is
nondecreasing in t. Proof. Let t + n = 2n d2 n te and t n = 2n d2 n te 1 and write v n (t) =
X In inf I< for all t 0. Proposition 2.2 A c`adl`ag function a : [0, ) R can be
da0 (I) + da00(I). (2.15) Summing over I n with sup I < t in (2.9), the terms in the sum telescope and we obtain v n (t) a 0 (t + n ) + a 00(t + n ). (2.16) Letting n
, the lefthand side converges to v(t) by definition, and the righthand side converge
and a := 1 2 (v a) are the smallest functions a 0 and a 00 with that property. Proof. Suppose
v(t) < for all t 0. Direction 1. Assume that a = a 0 a 00 for two cadlag nondecreasing
functions a 0 , a00, and let us show that v(t) < . This is the easy direct
Ftk = 0 . (3.9) Thus let t 0 and assume that tn t for simplicity. To compute E(H M) 2 t ), we expand the square and use the above orthogonality relation: E (H
M) 2 t = E nX1 k=0 Zk(Mtk+1Mtk ) 2 = nX1 k=0 E Z 2 k (Mtk+1Mtk ) 2 kHk 2 nX1 k=0 E (Mtk+1Mtk )
affine plane P = cfw_f(x0, y0) + ~ + ~. Then, for U a sufficiently small neighborhood of (x0,
y0), P will be tangent to the surface f(U). (b) Now assume ~ 6= 0 but ~ is proportional to ~.
Then ~ generates a line to which the surface f(U) will be tangent.
Brownian motion and its applications to path properties. We begin with the simple Markov
property, which takes a particularly nice form in this context. Theorem 1.10 Let (Bt , t 0) be a
Brownian motion, and let s > 0. Then (B t := Bt+s Bs, t 0) is a Brown
Xs = xs, s t = 0 for all t, and so the process (M f t )t0 is a martingale for all f. This property
will often serve to characterize (Xt)t0. Organization of the course. The course contains three
main parts of unequal size. In the first part we introduce an
dV (, s) < . (2.20) Then the process defined pathwise by (H A)t = Z (0,t] Hs dAs (2.21) is welldefined, c`adl`ag, adapted and of finite variation. Proof. First note
that the integral in (2.21)is welldefined for all t due to the finiteness of the integra
isomorphism V1 V2 and (V1 N1) = V2 N2. It follows that dim N1 = dim N2. This
allows us to set Definition 3.3.2. Let S be a submanifold and let a S. The dimension of S at a
is the dimension of the linear space N given in Definition 3.3.1. The codimension o