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26 Given a specification for the environment and the utility function, it seems like our problem is "just" one of coming up with a program that satisfies some specifications. It seems like you could go study that in software engineering. But, why not? Why is this not just software engineering? Any of us would be h...	https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf
to do to actual code to do it. But, it's a really hard problem and most people have given up on it. Unfortunately, it seems that's the problem we are faced with here. But, we're not going to do this automatically. So, what's the enterprise that we're going to be engaged in? We're going to look at classes of environ...	https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf
(P think? → • The table is too big! There are too many world states and too many sequences of percepts. Lecture 1 • 30 The problem is that the table is too big. If P is any size at all or if you live for very long, the table is way too big. Way, way too big. There are too many ways the world could be, there are t...	https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf
doesn't entertain alternative realities. There is certainly a class of problems for which you can't make a table but you can write a fairly compact program that would do the job of being the table. But there are other domains in which you quite clearly can't do that and those are the domains that we are going to fo...	https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf
osh" and then your brain would kick in and you'd start figuring out what to do about it. So, you could respond very flexibly to a very broad range of stimuli but there's no way that you could have stored your responses to them. 32 Learning • What if you don’t know much about the environment when you start or i...	https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf
lot about the world dynamics but I have to leave a free parameter representing this coefficient of friction. • Part of the agent’s job is to use sequences of percepts to estimate the missing details in the world dynamics. Lecture 1 • 34 Instead of giving the complete world dynamics; I'm going to have to leave a f...	https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf
no hard and fast distinction between learning and perceiving. 35 Classes of Environments Lecture 1 • 36 Let's think about environments and the different kinds of environments that our agents might need to work in. Now, a large part of this course will involve thinking about particular properties of the environme...	https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf
. Most things are not entirely deterministic, but some are reasonably well-modeled as being deterministic. In the first half of this course, we'll think about deterministic models of the environment, really as an abstraction, and in the second half we'll think about probabilistic models. 38 Classes of Environments...	https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf
.html) Backgammon is a game for two players, played on a board consisting of twenty- four narrow triangles called points. The triangles alternate in color and are grouped into four quadrants of six triangles each. The quadrants are referred to as a player's home board and outer board, and the opponent's home bo...	https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf
stone on the board. You could. But, that doesn't seem so useful. If you were building the robot to move the pieces, you would have to think of the x-y location; you would have to think of the motor voltages that you send to the joints in order for the arm to move where it needs to go in order to put the stone where...	https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf
Is it deterministic? No. There are two issues about backgammon that make it nondeterministic. One is the dice. The other is your opponent. Actually, games are an interesting special case of the model we’ve been exploring. There is a nice chapter in the book on games; but we're not going to cover it, just because th...	https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf
Example: Driving a Taxi Recitation Exercise: Think about how you would choose – • Action space – A? • Percept space – P? • Environment – E? Lecture 1 • 48 As an exercise for our next recitation, think about how you might model the problem of designing an automatic taxi driver. What would be an appropriate way t...	https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf
is accessible (you can see everything there is to see in one shot) this means that you don't need any memory; you can just react based on your current percepts. For example, consider the amount of gasoline in your car. There are (at least) two ways of knowing how much gas you have: one is to remember how much driv...	https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf
state a Policy • State estimator/Memory • What we’ve chosen to remember from the history of percepts • Maps what you knew before, what you just perceived and what you just did, into what you know now. • Problem of behavior: Given my mental state, what action should I take? Lecture 1 • 53 The second componen...	https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf
and imagine "what will happen if we take action1; what if we take action2”. Then, after we take action1, what if we take action2, etc. I just had a flat tire, what if I call AAA - then wait 6 hours. What if I fix it myself, probably get fixed in 1/2 hour but I'd get covered in mud. So, there are different consequen...	https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf
Type Inference and the Hindley-Milner Type System Armando Solar-Lezama Computer Science and Artificial Intelligence Laboratory M.I.T. With slides from Arvind. Used with permission. September 29, 2011 September 29, 2011 L07-1 Type Inference • Consider the following expression – (𝜆f:int int. f 5) (𝜆x:int. ...	https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/3805b29885701911f983fc8299f9d9ac_MIT6_820F15_L07.pdf
of Equality Constraints • Consider the following Language of Constraints • Constraints in this language have a lot of good properties – Nice and compositional – Linear time solution algorithm L06-6 Building Constraints from Typing Rules • Notation – The constraints on the right ensure that the judgmen...	https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/3805b29885701911f983fc8299f9d9ac_MIT6_820F15_L07.pdf
->  base types (Int, Bool ..) type variables Function types A substitution is a map S : Type Variables  Types S = [1 / t1,..., n / tn] ’ is a Substitution Instance of  ’ = S  Example: S = [(t -> Bool) / t1] S ( t1 -> t1) = Substitutions can be composed, i.e., S2 S1 Example: ( t -> Bool) -> ( ...	https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/3805b29885701911f983fc8299f9d9ac_MIT6_820F15_L07.pdf
otherwise = fail Does the order matter? No September 27, 2011 L06-11 Type inference strategy 2 • Like strategy 1, but we solve the constraints as we see them – Build the substitution map incrementally September 29, 2011 L06-12 Simple Inference Algorithm W(TE, e) returns (S,) such that S (...	https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/3805b29885701911f983fc8299f9d9ac_MIT6_820F15_L07.pdf
Case e of c x = = x.e = (e1 e2) = ({}, Typeof(c)) if (x  Dom(TE)) then Fail else let  = TE(x); in ({},  ) let (S1, 1) = W(TE + { x : u }, e) in (S1, S1(u) -> 1) let (S1, 1) = W(TE, e1); (S2, 2) = W(S1(TE), e2); u’s represent new type variables S3 = Unify(S2(1), 2 -> u); in (S3 S2 S1, S3...	https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/3805b29885701911f983fc8299f9d9ac_MIT6_820F15_L07.pdf
𝑊 𝑓: 𝑢0 , 𝑓 = (∅, 𝑢0) 𝑊 𝑓: 𝑢0 , 5 = (∅, 𝐼𝑛𝑡) 𝑈𝑛𝑖𝑓𝑦 𝑢0, 𝐼𝑛𝑡 → 𝑢1 = 𝑊 𝑓: 𝑢0 , 𝑓 5 = 𝑊 ∅, 𝜆𝑓. 𝑓 5 = 𝑊(∅, 𝜆𝑓. 𝑓 5 𝜆𝑥. 𝑥 ) Example def Unify(1, 2) = case (1, 2) of (1, t2) = [1 / t2] provided t2  FV(1) (t1, 2) = [2 / t1] provided t1  FV(2) (1, 2) = if (eq? 1...	https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/3805b29885701911f983fc8299f9d9ac_MIT6_820F15_L07.pdf
) 𝑢0 ] 𝑊 𝑓: 𝑢0 , 𝑓 5 = 𝑊 ∅, 𝜆𝑓. 𝑓 5 = 𝑊(∅, 𝜆𝑓. 𝑓 5 𝜆𝑥. 𝑥 ) Example Def W(TE, e) = Case e of … x.e = let (S1, 1) = W(TE + { x : u }, e) in (S1, S1(u) -> 1) (e1 e2) = let (S1, 1) = W(TE, e1); (S2, 2) = W(S1(TE), e2); S3 = Unify(S2(1), 2 -> u); in (S3 S2 S1, S3(u)) 𝑊 𝑓:...	https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/3805b29885701911f983fc8299f9d9ac_MIT6_820F15_L07.pdf
𝜆𝑥. 𝑥 ) Example Def W(TE, e) = Case e of … x.e = let (S1, 1) = W(TE + { x : u }, e) in (S1, S1(u) -> 1) (e1 e2) = let (S1, 1) = W(TE, e1); (S2, 2) = W(S1(TE), e2); S3 = Unify(S2(1), 2 -> u); in (S3 S2 S1, S3(u)) 𝑊 ∅, 𝜆𝑓. 𝑓 5 = ( (𝐼𝑛𝑡 → 𝑢1) 𝑢0 , 𝐼𝑛𝑡 → 𝑢1 → 𝑢1) 𝑊 ∅, 𝜆𝑥. ...	https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/3805b29885701911f983fc8299f9d9ac_MIT6_820F15_L07.pdf
2) then [ ] else fail (11->12, 21 ->22) = let S1=Unify(11, 21) S2=Unify(S1(12), S1(22)) in S2 S1 𝑈𝑛𝑖𝑓𝑦 𝐼𝑛𝑡 → 𝑢1 , 𝑢3 → 𝑢3 = 𝐼𝑛𝑡 𝑢3 ; 𝐼𝑛𝑡 𝑢1 𝑈𝑛𝑖𝑓𝑦 𝐼𝑛𝑡, 𝑢4 = [𝐼𝑛𝑡 𝑢4 ] 𝑈𝑛𝑖𝑓𝑦 𝐼𝑛𝑡 → 𝑢1 → 𝑢1, 𝑢3 → 𝑢3 → 𝑢4 = 𝐼𝑛𝑡 𝑢3 ; 𝐼𝑛𝑡 𝑢1 ; 𝐼𝑛𝑡/𝑢4 𝑊(∅, 𝜆𝑓....	https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/3805b29885701911f983fc8299f9d9ac_MIT6_820F15_L07.pdf
1, S3(u)) 𝑊 ∅, 𝜆𝑓. 𝑓 5 = ( (𝐼𝑛𝑡 → 𝑢1) 𝑢0 , 𝐼𝑛𝑡 → 𝑢1 → 𝑢1) 𝑊 ∅, 𝜆𝑥. 𝑥 = (∅, 𝑢3 → 𝑢3) 𝑈𝑛𝑖𝑓𝑦 𝐼𝑛𝑡 → 𝑢1 → 𝑢1, 𝑢3 → 𝑢3 → 𝑢4 = 𝐼𝑛𝑡 𝑢3 ; 𝐼𝑛𝑡 𝑢1 ; 𝐼𝑛𝑡/𝑢4 𝑊 ∅, 𝜆𝑓. 𝑓 5 𝜆𝑥. 𝑥 = ( (𝐼𝑛𝑡 → 𝑢1) 𝑢0 ; 𝐼𝑛𝑡 𝑢3 ; 𝐼𝑛𝑡 𝑢1 ; 𝐼𝑛𝑡/𝑢4 , 𝐼𝑛𝑡) What about Let? ...	https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/3805b29885701911f983fc8299f9d9ac_MIT6_820F15_L07.pdf
2 ∶ 𝜏 = ∃𝜏′, Γ; 𝑥: 𝜏′ ⊢ 𝑒1: 𝜏′ ∧ Γ; 𝑥: 𝜏′ ⊢ 𝑒2: 𝜏 • Algorithm Case Exp = let x = e1 in e2 => let (S1, 1) = W(TE + {x : u}, e1); S2 = Unify(S1(u), 1); (S3, 2) = W(S2 S1(TE) + {x : }, e2); in (S3 S2 S1, 2) September 29, 2011 L06-22 Polymorphism September 27, 2011 L06-23 Some obse...	https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/3805b29885701911f983fc8299f9d9ac_MIT6_820F15_L07.pdf
𝑡] • Example 𝑖𝑑 = Λ𝑇. 𝜆𝑥 ∶ 𝑇. 𝑥 𝑖𝑑 𝑖𝑛𝑡 5 September 27, 2011 L06-25 Different Styles of Polymorphism • Impredicative Polymorphism 𝜏 ::= 𝑏 𝜏1 → 𝜏2 𝑇 \| ∀𝑇. 𝜏 e ::= 𝑥 𝜆𝑥: 𝜏. 𝑒 𝑒1𝑒2 ΛT. e e[𝜏] • Very powerful – Although you still can’t express recursion • T...	https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/3805b29885701911f983fc8299f9d9ac_MIT6_820F15_L07.pdf
– (𝜆𝑠: ∀𝑇 . 𝜏 … 𝑠 𝑖𝑛𝑡 𝑥 … 𝑠 𝑏𝑜𝑜𝑙 𝑥)(Λ𝑇. 𝑐𝑜𝑑𝑒 𝑓𝑜𝑟 𝑠𝑜𝑟𝑡) L06-28 Let polymorphism • Introduce let x = e1 in e2 – Just like saying 𝜆𝑥. 𝑒2 𝑒1 – Except x can be polymorphic • Good engineering compromise – Enhance expressiveness – Preserve decidability • This is the Hindle...	https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/3805b29885701911f983fc8299f9d9ac_MIT6_820F15_L07.pdf
\| t \| 1 -> 2 Type Schemes  ::=  \| t.  base types type variables Function types Type Environments TE ::= Identifiers  Type Schemes Note, all the ’s occur in the beginning of a type scheme, i.e., a type  cannot contain a type scheme  September 29, 2011 L06-33 Instantiations  = t...	https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/3805b29885701911f983fc8299f9d9ac_MIT6_820F15_L07.pdf
��2 :𝜏′ Remember, 𝜏 stands for a monotype, 𝜎 for a polymorphic type (Abs) (Var) (Const) Γ ; 𝑥:𝜏 ⊢𝑒∶𝜏′ Γ⊢𝜆𝑥.𝑒∶𝜏→𝜏′ 𝑥:𝜎 ∈Γ 𝜎≥𝜏 Γ⊢𝑥:𝜏 𝑡𝑦𝑝𝑒𝑜𝑓(𝑐)≥𝜏 Γ⊢𝑐:𝜏 𝑥 can be considered of type 𝜏 as long as its type as specified in the environment can be specialized to 𝜏 (i.e. 𝜏 is an instance o...	https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/3805b29885701911f983fc8299f9d9ac_MIT6_820F15_L07.pdf
: u3.u3 -> u1} W(TE, (f 1) ) = ( [ ] , u1 ) ( [ ] , u4 -> u1 ) W(TE, f) = W(TE, 1) = ( [ ] , Int ) Unify(u4 -> u1 , Int -> u5) = ... [ Int / u4 , u1 / u5 ] September 29, 2011 L06-38 Important Observations • Do not generalize over type variables used elsewhere • Let is the only way of defining polymorph...	https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/3805b29885701911f983fc8299f9d9ac_MIT6_820F15_L07.pdf
Var) (Let) TE ├ e :  t  FV(TE) TE ├ e : t. TE ├ e : t. TE ├ e :  [u/t] (x : )  TE TE ├ x :  TE+{x:} ├ e1:  TE+{x:} ├ e2:’ TE ├ (let x = e1 in e2) : ’ (App) and (Abs) rules remain unchanged. Sound but no inference algorithm ! September 29, 2011 L06-43 MIT...	https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/3805b29885701911f983fc8299f9d9ac_MIT6_820F15_L07.pdf
Vectors Our very ﬁrst topic is unusual in that we will start with a brief written presentation. More typically we will begin each topic with a videotaped lecture by Professor Auroux and follow that with a brief written presentation. As we pointed out in the introduction, vectors will be used throughout the course....	https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/38223d6ec70b12de9d01446438100b9d_MIT18_02SC_notes_0.pdf
gure shows, this can be done in 11� either order B / / / / / A/ / / / / / / A + B / / / / / / / A / / / / / B It is often useful to think of vectors as displacements. In this way, A + B can be thought of as the displacement A followed by the displacement B. You subtract vectors eith...	https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/38223d6ec70b12de9d01446438100b9d_MIT18_02SC_notes_0.pdf
a1i + a2j. This is equal to the vector drawn from the origin to the point (a1, a2). 3. For A = a1i + a2j, a1 and a2 are called the i and j components of A. (Note that they are scalars.) −→ −−→ 5. P = OP is the vector from the origin to P . 22 � � y 6. On the blac...	https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/38223d6ec70b12de9d01446438100b9d_MIT18_02SC_notes_0.pdf
A − B a1 − b1 b2 − − −−→ → → For two points P and Q the vector PQ = Q − P i.e., PQ is the displacement from P to Q. −−→ y y O •Q = (q1, q2) −−→ PQ •P = (p1, p1) Q P x Vectors in three dimensions We represent a three dimensional vector as an arrow in space. Using coordinates we need three numbers to repres...	https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/38223d6ec70b12de9d01446438100b9d_MIT18_02SC_notes_0.pdf
:104)a1, a2, a3(cid:105) + (cid:104)b1, b2, b3(cid:105) = (cid:104)a1 + b1, a2 + b2, a3 + b3(cid:105). exactly as in the two dimensional case. Magnitude in three dimensions also follows from the Pythagorean theorem. \|a1i + a2j + a3k\| = \|(cid:104)a 1, a2, a3(cid:105)\| = (cid:113) a1 + a2 2 2 + a2 3 You can see this in t...	https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/38223d6ec70b12de9d01446438100b9d_MIT18_02SC_notes_0.pdf
(cid:104)3, 4(cid:105). 1 5 (cid:104)3, 4(cid:105) = (cid:28) (cid:29) 3 4 , 5 5 has unit length and is parallel to 4(cid:122) (cid:122) (cid:47) (cid:47) (cid:79) (cid:79) (cid:63) (cid:63) MIT OpenCourseWarehttp://ocw.mit.edu18.02SC Multivariable CalculusFall 2010For information about citing these materials or our T...	https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/38223d6ec70b12de9d01446438100b9d_MIT18_02SC_notes_0.pdf
6.837 Computer Graphics Curve Properties & Conversion, Surface Representations vectorportal.com 6.837 – Matusik 1 Cubic Bezier Splines • P(t) = (1-t)³ P1 + 3t(1-t)² P2 + 3t²(1-t) P3 + P4 t³ 2 Bernstein Polynomials • For Bézier curves, the basis polynomials/vectors are Bernstein polynomia...	https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/38682e92015a39734867853f57ccf55b_MIT6_837F12_Lec02.pdf
ifferential Properties of Curves • Motivation – Compute normal for surfaces – Compute velocity for animation – Analyze smoothness Image courtesy of Kristian Molhave on Wikimedia Commons. License: CC- BY-SA. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/hel...	https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/38682e92015a39734867853f57ccf55b_MIT6_837F12_Lec02.pdf
– K(t)=T’(t) – Magnitude \|\|K(t)\|\| is constant for a circle – Zero for a straight line • Always orthogonal to tangent, ie. 16Geometric Interpretation • K is zero for a line, constant for circle – What constant? 1/r • 1/\|\|K(t)\|\| is the radius of the circle that touches P(t) at t and has the same curvature as the...	https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/38682e92015a39734867853f57ccf55b_MIT6_837F12_Lec02.pdf
. 26 Cubic B-Splines • ≥ 4 control points • Locally cubic – Cubics chained together, again. Courtesy of Seth Teller. 27 Cubic B-Splines • ≥ 4 control points • Locally cubic – Cubics chained together, again. Courtesy of Seth Teller. 28 Cubic B-Splines • ≥ 4 control points • Locally cubic – Cubics chaine...	https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/38682e92015a39734867853f57ccf55b_MIT6_837F12_Lec02.pdf
control points as Bézier new Bézier control points to match B-Spline MIT EECS 6.837, Popović new BSpline control points to match Bézier original control points as B-Spline 37NURBS (Generalized B-Splines) • Rational cubics – Use homogeneous coordinates, just add w ! • Provides an extra weight para...	https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/38682e92015a39734867853f57ccf55b_MIT6_837F12_Lec02.pdf
(Note! We relabeled t to u) 43 How to Build Them? Here’s an Idea • P(u) = (1-u)³ P1 + 3u(1-u)² P2 + 3u²(1-u) P3 + u³ P4 (Note! We relabeled t to u) 44 How to Build Them? Here’s an Idea • P(u) = (1-u)³ P1 + 3u(1-u)² P2 + 3u²(1-u) P3 + u³ P4 (Note! We relabeled t to u) 45 ...	https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/38682e92015a39734867853f57ccf55b_MIT6_837F12_Lec02.pdf
P4(v) + u³ • Let’s make the Pis move along curves! v=0 v=1/3 v=1 49 Here’s an Idea • P(u, v) = (1-u)³ P1(v) + 3u(1-u)² P2(v) + 3u²(1-u) P3(v) P4(v) + u³ • Let’s make the Pis move along curves! v=0 v=1/3 v=2/3 v=1 50 Here’s an Idea • P(u, v) = (1-u)³ P1(v) + 3u(1-u)²...	https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/38682e92015a39734867853f57ccf55b_MIT6_837F12_Lec02.pdf
control points Pi,j 16 2D basis functions Bi,j 59 Recap: Tensor Bézier Patches • Parametric surface P(u,v) is a bicubic polynomial of two variables u & v • Defined by 4x4=16 control points P1,1, P1,2.... P4,4 • Interpolates 4 corners, approximates others • Basis are product of two Bernstein polynomial...	https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/38682e92015a39734867853f57ccf55b_MIT6_837F12_Lec02.pdf
of surface at (u,v) Column vector of basis functions (v) Row vector of basis functions (u) 4x4 matrix of x coordinates of the control points 66Hardcore: Matrix Notation for Patches • Curves: • Surfaces: A separate 4x4 geometry matrix for x, y, z • T = power basis B = spline matrix G = geometry matrix 6...	https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/38682e92015a39734867853f57ccf55b_MIT6_837F12_Lec02.pdf
Durand 71Cool: Displacement Mapping • Not all surfaces are smooth... © Addison-Wesley. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/. 72Cool: Displacement Mapping • Not all surfaces are smooth... • “Paint” displace...	https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/38682e92015a39734867853f57ccf55b_MIT6_837F12_Lec02.pdf
This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/. 88 Subdivision Curves and Surfaces • Advantages . l a t e n e r r a W © IEEE. All rights reserved. This content is excluded from our Creative Commons license. For more information, see ...	https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/38682e92015a39734867853f57ccf55b_MIT6_837F12_Lec02.pdf
2nd dimension v s(u,v) s(u,v)=R(v)q(u) where R is a matrix, q a vector, and s is a point on the surface 94 General Swept Surfaces • Trace out surface by moving a profile curve along a trajectory. – profile curve q(u) provides one dim – trajectory c(u) provides the other • Surface of revolution can be see...	https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/38682e92015a39734867853f57ccf55b_MIT6_837F12_Lec02.pdf
given by tangent of profile curve, the other by tangent of trajectory s(u,v)=M(c(v))q(u) where M is a matrix that depends on the trajectory c 100 Questions? 101Implicit Surfaces • Surface defined implicitly by a function This image is in the public domain. Source: Wikimedia Commons. 102Implicit Surfaces • ...	https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/38682e92015a39734867853f57ccf55b_MIT6_837F12_Lec02.pdf
107 Questions? 108That’s All for Today • Further reading – Buss, Chapters 7 & 8 • Subvision curves and surfaces – http://www.cs.nyu.edu/~dzorin/sig00course/ 6.837 – Durand 0 9 109 MIT OpenCourseWare http://ocw.mit.edu 6.837 Computer Graphics Fall 2012 For information about citing these materi...	https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/38682e92015a39734867853f57ccf55b_MIT6_837F12_Lec02.pdf
3.225 Electrical,Optical, and Magnetic Properties of Materials • Professor Eugene Fitzgerald • Purpose: connect atoms and structure to properties • Semi-historical context – What was understood first from the micro to the macro? – What was missing to explain other materials? Origin of Conduction Range of Resi...	https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/386e36fc0277bd2f6cb8e49e14d0882d_lecture_1.pdf
σE In general, r ~ r J = E σ All material info In cubic material, Isotropic material E Anisotropic material E J J J ⎛ ⎜ J ⎜ y ⎜ J z ⎝ ⎞ x ⎟ ⎟ ⎟ ⎠ = σ ⎛ ⎜ σ ⎜ ⎜ σ ⎝ xx xy xz σ σ σ xy yy yz σ σ σ ⎞ E ⎛ xz ⎟ ⎜ E ⎟ ⎜ yz ⎜ ⎟ ⎠ E ⎝ zz ⎞ x ⎟ ⎟ y ⎟ ⎠ z Microscopic Origin: Can we predict Conductivity ...	https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/386e36fc0277bd2f6cb8e49e14d0882d_lecture_1.pdf
p t( ) τ + F t( ) + F ( )t +... 2 1 Response (ma) Drag Driving Force Restoring Force... dp t ( ) dt ≈ − p t( ) τ − eE Add a drag term, i.e. the electrons have many collisions during drift 1/τ represents a ‘viscosity’ in mechanical terms In steady state, dp t ( ) dt = 0 −t p t( ) = p∞ (1− e τ ) p∞ = −...	https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/386e36fc0277bd2f6cb8e49e14d0882d_lecture_1.pdf
Sr Ba Nb Fe Mn (a) Zn Cd Hg (78 K) Al Ga In Tl Sn Pb Bi Sb 1 1 1 1 1 1 1 1 2 2 2 2 2 1 2 2 2 2 2 3 3 3 3 4 4 5 5 4.70 2.65 1.40 1.15 0.91 8.47 5.86 5.90 24.7 8.61 4.61 3.55 3.15 5.56 17.0 16.5 13.2 9.27 8.65 18.1 15.4 11.5 10.5 14.8 13.2 14.1 16.5 o rs(A) 1.72 2.08 2.57 2.75 2.98 1.41 1.60 1.59 0.99 1.41 1.73 1.89 1.96...	https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/386e36fc0277bd2f6cb8e49e14d0882d_lecture_1.pdf
Cu Ag Au Be Mg Ca Sr Ba Nb Fe Zn Cd Hg Al Ga In Tl Sn Pb Bi Sb 7.3 17 18 14 8.6 21 20 12 6.7 1.4 0.66 2.1 3.2 2.4 2.4 0.71 6.5 0.84 1.7 0.91 1.1 0.57 0.072 0.27 0.88 3.2 4.1 2.8 2.1 2.7 4.0 3.0 0.51 1.1 2.2 0.44 0.19 0.42 0.24 0.49 0.56 0.80 0.17 0.38 0.22 0.23 0.14 0.023 0.055 373 K 0.61 1.9 2.8 2.1 0.27 0.74 1.5 0.33...	https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/386e36fc0277bd2f6cb8e49e14d0882d_lecture_1.pdf
, deformed dislocations 2x10-8 Cu-1%Sn Cu 0 100 200 300 solute atoms lattice vibrations Temperature (K) Example: Conductivity Engineering • Objective: increase strength of Cu but keep conductivity high σ = ne2τ m l = vτ μ= eτ m Scattering length connects scattering time to microstructure Dislo...	https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/386e36fc0277bd2f6cb8e49e14d0882d_lecture_1.pdf
Today's plan: [Any questions about lock server lab?] Reviewing event driven programming Outline structure of the remaining labs Common libasync/libarpc/nfsloop programming idioms: writing rpc client code writing async functions that call RPCs writing rpc server code Flash Event driven programming Ach...	https://ocw.mit.edu/courses/6-824-distributed-computer-systems-engineering-spring-2006/38dc024fe5e0b8fe7115d4e19d2476ba_lec3_events.pdf
write servers (asrv/svccb) Asynchronous RPC: needs a callback! [Example 3] Note: Cite as: Robert Morris, course materials for 6.824 Distributed Computer Systems Engineering, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. ...	https://ocw.mit.edu/courses/6-824-distributed-computer-systems-engineering-spring-2006/38dc024fe5e0b8fe7115d4e19d2476ba_lec3_events.pdf
IM to wife, research ticket prices, reply Or Amazon.com login... [Example 6] An aside on locking: No locking etc needed usually: e.g. to increment a variable When do you need locking? When an operation involving multiple stages Be careful about callbacks that are supposed to happen "later" e.g. delaycb (...	https://ocw.mit.edu/courses/6-824-distributed-computer-systems-engineering-spring-2006/38dc024fe5e0b8fe7115d4e19d2476ba_lec3_events.pdf
MASSACHUSETTS INSTITUTE OF TECHNOLOGY SLOAN SCHOOL OF MANAGEMENT 15.565 Integrating Information Systems: Technology, Strategy, and Organizational Factors 15.578 Global Information Systems: Communications & Connectivity Among Information Systems Spring 2002 Lecture 11 EMERGING TECHNOLOGIES “THE LAST MILE” (xDSL / C...	https://ocw.mit.edu/courses/15-565j-integrating-esystems-global-information-systems-spring-2002/3903d757ddd00703e1d0fa181da78ae6_lecture11.pdf
A B C SUBSCRIBERS NETWORK SOURCE • HOW SIMILAR TO AND DIFFER FROM ETHERNET – IMPACT ON PERFORMANCE / PRIVACY • WHY ASYMMETRIC (NORMALLY) • DIRECTION OPTIONS: – NO UPSTREAM (UNI-DIRECTION) – LIMITED UPSTREAM (ASYMMETRIC) – SYMMETRIC • ROLE OF SATELLITE TV AND TELCO’s 5 xDSL MODEMS • DSL = DIGITAL SUBSCRIBE...	https://ocw.mit.edu/courses/15-565j-integrating-esystems-global-information-systems-spring-2002/3903d757ddd00703e1d0fa181da78ae6_lecture11.pdf
(with differing needs) • Internet / Intranet Access • Web browsing vs hosting • E-mail • Remote LAN • Video conferencing • Transaction processing • IP telephony • Call center services • Video telephony (video conference) • High-definition TV • Video-on-demand • Leased line backup 8	https://ocw.mit.edu/courses/15-565j-integrating-esystems-global-information-systems-spring-2002/3903d757ddd00703e1d0fa181da78ae6_lecture11.pdf
In this lecture, we continue (and finish) the theme of geometric data structures. First, we'll finish our coverage of fractional cascading from Lecture 3 by illustrating a really cool application: 3D orthogonal range searching in O(log n) time. This gives us the second log factor improvement mentioned in Lecture 3, ...	https://ocw.mit.edu/courses/6-851-advanced-data-structures-spring-2012/396340161b7bd4c5cfc8820210a9743a_MIT6_851S12_Lecture4.pdf
6.826—Principles of Computer Systems 2002 6.826—Principles of Computer Systems 2002 9. Atomic Semantics of Spec This handout defines the semantics of the atomic part of the Spec language fairly carefully. It tries to be precise about all difficult points, but is sloppy about some things that seem obvious in orde...	https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/39afa4d7545639c4a282d63b8f9978a0_9.pdf
. In general, descriptions of programming languages are not in that state of grace. If you read the Pascal manual or the C manual carefully you will come away with a number of questions about exactly what happens if I do this and this, questions which the manual will not answer adequately. Two reasonably intelligen...	https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/39afa4d7545639c4a282d63b8f9978a0_9.pdf
We are going to get down to the foundations of Spec, and we are going to see another, very different application of Spec, a programming language rather than a file system. For this lecture, we will only talk about the sequential or atomic semantics of Spec, not about concurrent semantics. Consider the program: thr...	https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/39afa4d7545639c4a282d63b8f9978a0_9.pdf
ways to put things together in Spec: A , B A ; B a + b A [] B State What are the meanings going to be? Our basic notion is that what we are doing when writing a Spec program is describing a state machine. The central properties of a state machine are that it has states and it has transitions. A state is a func...	https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/39afa4d7545639c4a282d63b8f9978a0_9.pdf
variables into these two part names, so that each name refers to exactly one thing. This transformation makes things simpler to describe and understand, but uglier to read. It doesn’t change the meaning of the program, which could have been written with two part names in the first place. All the global variables ha...	https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/39afa4d7545639c4a282d63b8f9978a0_9.pdf
antics of Spec 4 6.826—Principles of Computer Systems 2002 6.826—Principles of Computer Systems 2002 There are three types of expressions: Type constant variable Example Meaning 1 x (\ s \| 1) (\ s \| s("x")) function invocation f(x) next sub-section (The type of these lambda’s is not quite right, as w...	https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/39afa4d7545639c4a282d63b8f9978a0_9.pdf
... s("f") ... s("x") ... How are we going to put it together, remembering the type we want for f(x), which is S -> U? f(x) means (\ s \| s("f") (s("x"), s)) ) Now this could be complete nonsense, for instance if s("f") evaluates to an integer. If s("f") isn’t a function then this doesn’t typecheck. But there is no ...	https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/39afa4d7545639c4a282d63b8f9978a0_9.pdf
way of coding that an exception has happened. Again, there are many ways to code that. The way we use is that an outcome has the same type as a state: it’s a function from names to values. However, there are a couple of funny names that you can’t actually write in the program. One of them is $x, and we adopt the con...	https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/39afa4d7545639c4a282d63b8f9978a0_9.pdf
have to say that you don’t mess around with the rest of the state. The way you do that is to say that the outcome is equal to the state except at the variable. o = s{"x" -> ME(e)(s)} This is just a Spec function constructor, of the form f{arg -> value}. Note that we are using the semantics of expressions that we de...	https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/39afa4d7545639c4a282d63b8f9978a0_9.pdf
is that p is an abstraction of a state transition, so its meaning will be a relation of type ATr. What about the argument x? There are many ways to deal with it. Our way is to use another pseudo- variable $a to pass the argument and get back the result. The meaning of p(e) is going to be (\ s, o \| (s {"$a" -> ME(...	https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/39afa4d7545639c4a282d63b8f9978a0_9.pdf
this won’t typecheck). VAR s' := s{"$a" -> ME(e)(s)} \| RET ME(p)(s) (s',o) are two ways of writing exactly the same thing, this RET ME(p)(s)(ME(e)(s)) (s,o) Aside—an alternate encoding for invocation Here is a different way of communicating the argument value to the function; you can skip this section if you like...	https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/39afa4d7545639c4a282d63b8f9978a0_9.pdf
supply, ME(e)(s) is the argument that picks out a particular function, to which we finally pass (s, o). However, there are lots of other ways to do this. One of the things which makes the semantics game hard is that there are many choices you can make. They don’t really make that much difference, but they can creat...	https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/39afa4d7545639c4a282d63b8f9978a0_9.pdf
unct s("$havoc"), so that if $havoc is true in s, any command relates s to any o. Now for the compound commands. Commands — c1 [] c2 MC(c1) MC(c2) (s, o) (s, o) \/ or on one line, MC(c1)(s, o) \/ MC(c2)(s, o) Non-deterministic choice is the ‘or’ of the relations. Commands — c1 [*] c2 It is clear we should ...	https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/39afa4d7545639c4a282d63b8f9978a0_9.pdf
, it requires that our semantics move through an intermediate state. If these were functions (if we could describe the commands as functions) then we could simply describe a sequential composition as (F2 (Fl s)). However, because Spec is not a functional language, we need to compose relations, in other words, to est...	https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/39afa4d7545639c4a282d63b8f9978a0_9.pdf
of the composition and we ignore c2. (EXISTS o' \| MC(cl)(s, o') /\ ( ~IsX(o') /\ MC(c2) \/ IsX(o') /\ o' = o)) Commands — c1 EXCEPT xs => c2 Now, what if we have a handler for the exception? If we assume (for simplicity) that all exceptions are handled, we simply have the complement of the semicolon case. If ther...	https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/39afa4d7545639c4a282d63b8f9978a0_9.pdf
Atomic Semantics of Spec 12 6.826—Principles of Computer Systems 2002 6.826—Principles of Computer Systems 2002 Overview Terminology The semantics of Spec are defined in three stages: expressions, atomic commands, and non- atomic commands (treated in handout 17 on formal concurrency). For the first two there i...	https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/39afa4d7545639c4a282d63b8f9978a0_9.pdf
notation. Also, it is instructive to see how the task is writing this particular kind of spec can be handled in Spec. You might wonder how this spec is related to code for Spec, that is, to a compiler or interpreter. It does look a lot like an interpreter. As with other specs written in Spec, however, this one is no...	https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/39afa4d7545639c4a282d63b8f9978a0_9.pdf
, typeX}) % eXception = SET X = Name -> V WITH {isX:=OIsX} = O SUCHTHAT (\ o \| ~ o.isX) = (S, O) -> Bool % eXception Set % Outcome % State % Atomic Transition X XS O S ATr CONST letter digit ids globals noX retX loopX typeX trueV := "ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz".rng :=...	https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/39afa4d7545639c4a282d63b8f9978a0_9.pdf
13 Handout 9. Atomic Semantics of Spec 14 6.826—Principles of Computer Systems 2002 6.826—Principles of Computer Systems 2002 The meaning of an id or var is just the string, of an exceptionSet the set of strings that are the exceptions in the set, of a type the set of values of the type. For the other constr...	https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/39afa4d7545639c4a282d63b8f9978a0_9.pdf
can’t be done statically. To take account of this and to ensure that the meaning of expressions is independent of the static type checking, we assume that in the context var := e the expression e is replaced by e AS t, where t is the declared type of var. The meaning of e AS t in state s is ME(e)(s) if that is in t...	https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/39afa4d7545639c4a282d63b8f9978a0_9.pdf
) MC(c2)(s, o) MC(c1)(s, o) \/ MC(c1)(s, o) \/ ( MC(c2)(s, o) /\ ~(EXISTS o' \| MC(c1)(s, o')) ) \/ ( EXISTS o' \| MC(c1)(s, o) /\ o .isX MC(c1)(s, o') /\ ~ o'.isX /\ MC(c2)(o',o ) ) c1 EXCEPT xs => c2 \/ ( EXISTS o' \| MC(c1)(s, o) /\ ~ o ("$x") IN xs MC(c1)(s, o') /\ o'("$x") IN xs /\ MC(c2)(o'{"$x" -> noX}...	https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/39afa4d7545639c4a282d63b8f9978a0_9.pdf
an ATr (atomic transition) relation. Thus the meaning of commands is given by a function MC with type C->ATr, where ATr = (S, O) -> Bool. We can define the ATr relation for each command by a predicate: a command relates state s to outcome o iff the predicate on s and o is true. We give the predicates in table 1 and...	https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/39afa4d7545639c4a282d63b8f9978a0_9.pdf
of these that don’t involve an invocation of an APROC are deterministic; in other words, the relation is a function. Furthermore, they are all total unless they involve an invocation that is partial. A RET produces the exception retX and leaves the returned value in $a. A RAISE yields an exceptional outcome which r...	https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/39afa4d7545639c4a282d63b8f9978a0_9.pdf
is the same as o except that id is undefined in o. It is this existential quantifier that makes the spec useless as an interpreter for Spec. << ... >>, IF ... FI or BEGIN ... END brackets don’t affect MC. The meaning of DO c OD can’t be given so easily. It is the fixed point of the sequence of longer and longer rep...	https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/39afa4d7545639c4a282d63b8f9978a0_9.pdf
type of the formal. 2. Remove local names from the state, since a routine shares only global state with its invoker. 3. Bind the value to the formal. 4. Find out using MC how the routine body relates the resulting state to an outcome. 5. Make the invoker's outcome from the invoker’s local state and the routine's fi...	https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/39afa4d7545639c4a282d63b8f9978a0_9.pdf
the routine didn’t supply a result of the correct type. An exception raised in the body is passed on. FUNC MR(r) -> (S->ATr) = VAR id1, id2, t1, t2, xs, c0 \| r = R« APROC id1(id2: t1)->t2 RAISES xs = << c0 >> » \/ r = R« FUNC id1(id2: t1)->t2 RAISES xs = RET (\ static: S \| (\ s, o \| c0 » => s("$a") IN t1 /\ (...	https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/39afa4d7545639c4a282d63b8f9978a0_9.pdf
and it is the value of ME((LAMBDA ...)). The value of (LAMBDA ...), like the value of any expression, is the result of evaluating ME((LAMBDA ...)) on the current state. This yields a transition function as we expect, and that function captures the local state of the LAMBDA expression; this is standard static scoping...	https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/39afa4d7545639c4a282d63b8f9978a0_9.pdf
3.15 Displays C.A. Ross, DMSE, MIT References: Braithwaite and Weaver chapter 6.4 Displays can work by -emitting light, eg. Incandescence (light from heating a filament) Cathodoluminescence (light from electron bombardment) ZnS +Ag (blue), (Zn,Cd)S +Cu (green), Y2O2S +Eu,Tb (red) –excited by three 20kV electron gun...	https://ocw.mit.edu/courses/3-15-electrical-optical-magnetic-materials-and-devices-fall-2006/39b18424818b1face2ad6579a915e34b_lecture14.pdf
display is dark. Colors come from dyes or filters. Response time limited by viscosity. Electrode patterns are made from thin film transistors (TFTs) made on glass – these use a-Si to make the channel region, with p-type source and drain doped with Al, and a Cr gate. The drain is connected to a pixel made from ITO (indi...	https://ocw.mit.edu/courses/3-15-electrical-optical-magnetic-materials-and-devices-fall-2006/39b18424818b1face2ad6579a915e34b_lecture14.pdf
Genesis of Friction between Macroscale contacts • Reference: Chapter 3 of the text books What is friction? F = ∂W ∂s µ varies as a function of the sliding distance. 1 0.6 0.2 0 0 20 40 60 80 D i stan ce slid ( m ) Scale issues in tribology Table 3.1 Scales in Tribology and Typical alues Adapte...	https://ocw.mit.edu/courses/2-800-tribology-fall-2004/39bee0a6dd75b805338d29c25b8a0810_ch3_friction.pdf
Si wafer Si-DLC a-C:H Sputtered DLC Optimized DLC 1 1 1 1 1 1 10 10 10 10 10 10 100 100 100 100 100 100 Relative friction forces in MEMS of two flat and smooth surfaces as functions of the distance between the two surfaces 103 100 h-1 Capillary at 45% RH van der Walls Electrostatic ) 2 m m N m ( a e r a t ...	https://ocw.mit.edu/courses/2-800-tribology-fall-2004/39bee0a6dd75b805338d29c25b8a0810_ch3_friction.pdf
1020 steel on iron, 1020, 1045, and 1095 steel (I) to (l) -- 1045 steel on iron, 1020, 1045, and 1095 steel (m) to (p) -- 1095 steel on iron, 1020, 1045, and 1095 steel Photos removed for copyright reasons. See Figure 3.1 in [Suh 1986]: Suh, N. P. Prentice-Hall, 1986. ISBN: 0139309837. Tribophysics. Englewood Cli...	https://ocw.mit.edu/courses/2-800-tribology-fall-2004/39bee0a6dd75b805338d29c25b8a0810_ch3_friction.pdf
Experimental conditions Experimental conditions Speed 1 mm/sec Normal load Temperature 1, 5 gf 25±2 ℃ Humidity Distance 35, 50, 70 % 2.4 m Pin-on-reciprocator tester Experimental setup Specimens • Pin specimens - Bearing ball (1/16”) - Slider (Nano type) • Flat specimens - µ-structured Si(coated) : L...	https://ocw.mit.edu/courses/2-800-tribology-fall-2004/39bee0a6dd75b805338d29c25b8a0810_ch3_friction.pdf
particles Re-input particles t n e i c i f f e o C n o i t c i r F 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.5 1.0 1.5 2.0 2.5 Sliding distance(m) Six stages in the frictional force versus distance slid relationship µ I II III IV V VI Distance slid Hard stationary surface polished by a soft surface Mirror finish ...	https://ocw.mit.edu/courses/2-800-tribology-fall-2004/39bee0a6dd75b805338d29c25b8a0810_ch3_friction.pdf
0 200 400 600 Cycles 800 1000 1200 Friction in Geometrically Confined Space 80 70 60 50 40 30 20 10 0 F1 N1 µ1 Transition 2.5 2 1.5 1 0.5 0 200 400 600 Cycles 800 1000 0 1200 Friction in Geometrically Confined Space Bearing Shaft 125 Dimensions in µ m Friction in Geometrically Confined Space 2.5...	https://ocw.mit.edu/courses/2-800-tribology-fall-2004/39bee0a6dd75b805338d29c25b8a0810_ch3_friction.pdf

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