text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
(cid:2)(cid:4)(cid:3)
(cid:2)(cid:6)(cid:3)
(cid:21)(cid:20)
(cid:23)(cid:22)
(4.129)
(cid:23)(cid:22)
(cid:23)(cid:20)
(cid:8) , and similarly for the other functions. This quadrature
where (cid:23)
(cid:8) .
scheme is exact for linear variations of the kernel (cid:21)
In the present case (cid:24)
(cid:17)(cid:20)(cid... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
:17)(cid:20)(cid:19)
(cid:4)(cid:2)
(cid:17)(cid:20)(cid:19)
(cid:4)(cid:2)
(4.131)
(cid:17)(cid:9)(cid:19)
Here it is interesting to note that Eq. (4.130) is identical to Eq. (4.108) except for the
simple change in integration weight from (cid:23)
, basically applying a sinc-function
squared to the field amplitude vs r... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
)
"
$
(cid:13)
(cid:2)
(cid:21)
(cid:3)
(cid:8)
(cid:28)
(cid:5)
(cid:1)
(cid:9)
(cid:11)
(cid:21)
(cid:3)
(cid:8)
(cid:28)
(cid:5)
(cid:0)
(cid:9)
(cid:16)
(cid:6)
(cid:23)
(cid:2)
(cid:10)
$
(cid:6)
(cid:21)
(cid:10)
(cid:21)
(cid:3)
(cid:8)
(cid:25)
(cid:21)
(cid:3)
(cid:10)
(cid:9)
(cid:8)
(cid:10)
(cid:6)
(cid:10)... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
ities are usually the controlling factor, and since the number of
singularities increase with frequency, and since singularities are not necessarily bet-
ter represented by a linear than a constant kernel, the improvement in computational
efficiency is much less pronounced for underwater acoustic problems, and the Filon... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
be rather insignificant. Further, for computation of
transmission loss, usually performed on a dense spatial grid, the fact that the adaptive
sampling has to be performed individually for each receiver makes it rather inten-
sive computationally. However, for time domain computations for a small number of
receivers, it ... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
Lecture 1
Overview of some probability
distributions.
In this lecture we will review several common distributions that will be used often throughtout
the class. Each distribution is usually described by its probability function (p.f.) in the case
of discrete distributions or probability density function (p.d.f.) i... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/46afee1cbe70839c128ef68b2a6b1e16_lecture1.pdf |
is defined by
∞
R we have P(X = a) = 0. Given a function � :
R, the
X �
E�(X) =
�
�(x)p(x)dx.
�
−�
Notation. The fact that a random variable X has distribution P will be denoted by
P.
Normal (Gaussian) Distribution N(�, π2). Normal distribution is a continuous dis
X
�
tribution on R with p.d.f.
p(x) =
1
≤... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/46afee1cbe70839c128ef68b2a6b1e16_lecture1.pdf |
y
1
≤2ϕ
2 y
2 dy = 0
e−
since the integrand is an odd function. To compute the second moment EY 2 , let us first note
1
is a probability density function, it integrates to 1, i.e.
that since �
2
y
2
2� e−
If we integrate this by parts, we get,
�
−�
�
1 =
1
≤2ϕ
2 y
dy.
2
e−
1 =
�
2
y
2
dy =
e−
1
≤2ϕ... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/46afee1cbe70839c128ef68b2a6b1e16_lecture1.pdf |
�i, π2) then their sum will also have a normal distribution
Xi, 1
i
n, such that Xi �
≈
≈
X1 + . . . + Xn �
N(�1 + . . . + �n, π1
2 + . . . + π2 ).n
Normal distribution appears in one of the most important results that one learns in probabil
ity class, namely, a Central Limit Theorem (CLT), which states the foll... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/46afee1cbe70839c128ef68b2a6b1e16_lecture1.pdf |
=
X
p(1) = P(X = 1) = p, p(0) = P(X = 0) = 1
−
p for some p
[0, 1].
∞
It is easy to check that
EX = p, Var(X) = p(1
−
p).
Binomial Distribution B(n, p). This distribution describes a random variable X that
is a number of successes in n trials with probability of success p. In other words, X is a
sum of n in... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/46afee1cbe70839c128ef68b2a6b1e16_lecture1.pdf |
that X will exceed level t + s given that it
will exceed level t can be computed as follows:
P(X
t + s X
|
∼
∼
t) =
P(X
t + s, X
∼
P(X
t)
∼
�t = e−
�(t+s)/e−
∼
= e−
t)
=
P(X
t + s)
t)
∼
P(X
�s = P(X
∼
s),
∼
i.e.
P(X
X
t + s
|
t) = P(X
s).
∼
∼
If X represent a lifetime of some object in s... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/46afee1cbe70839c128ef68b2a6b1e16_lecture1.pdf |
phone number; distribution of bacteria on some surface or weed
in the field. All these examples share some common properties that give rise to a Poisson
distribution. Suppose that we count a number of random objects in a certain region T and
this counting process has the following properties:
4
1. Average nu... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/46afee1cbe70839c128ef68b2a6b1e16_lecture1.pdf |
0 + P(Xi = 1) + αn,
0
k
�
�
k
2 kP(Xi = k), and by the last property above we assume that αn becomes
where αn =
small with n, since the probability to observe more that two objects on the interval of size T /n
becomes small as n becomes large. Combining two equations above gives, P(Xi = 1)
� T . n
2 is small, ... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/46afee1cbe70839c128ef68b2a6b1e16_lecture1.pdf |
associated with this distribution. For example,
’normrnd’ generates random numbers from distribution ’norm’, ’normpdf’ gives p.d.f., ’norm
cdf’ gives c.d.f., ’normfit’ fits the normal distribution for a given dataset (we will look at
this last type of functions when we discuss Maximum Likelihood Estimators). Please, l... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/46afee1cbe70839c128ef68b2a6b1e16_lecture1.pdf |
MIT 6.972 Algebraic techniques and semidefinite optimization
February 28, 2006
Lecturer: Pablo A. Parrilo
Scribe: ???
Lecture 6
Last week we learned about explicit conditions to determine the number of real roots of a univariate
polynomial. Today we will expand on these themes, and study two mathematical objects o... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/46bf636ecc6195991e396703bfe1a1ae_lecture_06.pdf |
.
⎢
⎢
⎢
p(x0)x0
⎢
⎢
p(x0)
⎢
⎢
q(x0)x0
⎢
⎢
q(x0)x0
⎢
⎢
. . .
⎢
⎢
⎢
⎣
q(x0)x0
q(x0)
n−2
n−1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
= 0.
This implies that the matrix on the lefthand side, called the Sylvester matrix Sylx(p, q) associated
to p and q, is singular and thus its determinant must vanish. It is not too... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/46bf636ecc6195991e396703bfe1a1ae_lecture_06.pdf |
q(αj ) = p det q(Cp)
n
j=1 k=1
j=1
= (−1)nm q
n
m
m
�
p(βk ) = (−1)nm q det p(Cq )
n
m
k=1
(1)
• Kronecker products: Using a wellknown connection to Kronecker products, we can also write
(1) as
•
B´ezout matrix
To be completed
m n
pn q det(Cp ⊗ Im − I
m
n ⊗ Cq ).
ToDo
If can be shown that all these const... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/46bf636ecc6195991e396703bfe1a1ae_lecture_06.pdf |
easily checkable condition for the simultaneous
vanishing of two univariate polynomials. Can we use the resultant to produce a condition for a polynomial
to have a double root? Recall that if a polynomial p(x) as a double root at x0 (which can be real or
complex), then its derivative p�(x) also vanishes at x0. Thus,... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/46bf636ecc6195991e396703bfe1a1ae_lecture_06.pdf |
3.1 Polynomial equations
One of the most natural applications of resultants is in the solution of polynomial equations in two
variables. For this, consider a polynomial system
p(x, y) = 0,
q(x, y) = 0,
(2)
with only a finite number of solutions (which is generically the case). Consider a fixed value of y0, and
the ... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/46bf636ecc6195991e396703bfe1a1ae_lecture_06.pdf |
the corresponding value of x� ≈ −1.3853.
3.2
Implicitization of rational curves
To be completed
3.3 Random matrices
To be completed
ToDo
ToDo
4
The set of nonnegative polynomials
One of the main reasons why nonnegativity conditions about polynomials are difficult is because these
sets can have a quite complicate... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/46bf636ecc6195991e396703bfe1a1ae_lecture_06.pdf |
but it also vanishes at points inside the set.
Why is this?
Example 5. Consider the univariate polynomial p(x) = x4 + 2ax2 + b. For what values of a, b does it
hold that p(x) ≥ 0 ∀x ∈ R? Since the leading term x4 has even degree and is strictly positive, p(x) is
strictly positive if and only if it has no real roots... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/46bf636ecc6195991e396703bfe1a1ae_lecture_06.pdf |
511.52b0042-2-1.5-1-0.50.511.52a-1-0.50.511.52b-3-2-1123a-1123b0220444-3-2-1123a-1123b Figure 3: A threedimensional convex set, described by one quadratic and one linear inequality, whose
projection on the (a, b) plane is equal to the set in Figure 1.
One has to do with its algebraic structure, and the other one wit... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/46bf636ecc6195991e396703bfe1a1ae_lecture_06.pdf |
5 and Figure 1.
The presence of “extraneous” components of the discriminant inside the feasible set is an important
roadblock for the availability of “easily computable” barrier functions. Indeed, every polynomial that
vanishes on the boundary of the set Pn must necessarily have the discriminant as a factor. This is... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/46bf636ecc6195991e396703bfe1a1ae_lecture_06.pdf |
F ⊆ S, with the property that x, y ∈ S, 1
F is exposed if it can be written as F = S ∩ H, where H is a supporting hyperplane of S.
2 (x + y) ∈ F ⇒ x, y ∈ F . A face
65
−20246024012345abt Figure 4: The discriminant of the polynomial x4 + 4ax3 + 6bx2 + 4cx + 1. The convex set inside the
“bowl” corresponds to the re... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/46bf636ecc6195991e396703bfe1a1ae_lecture_06.pdf |
In
Proceedings of the American Control Conference, 2002.
[RG95] M. Ramana and A. J. Goldman. Some geometric results in semidefinite programming. J.
Global Optim., 7(1):33–50, 1995.
[Stu98] B. Sturmfels. Introduction to resultants. In Applications of computational algebraic geometry
(San Diego, CA, 1997), volume 53 ... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/46bf636ecc6195991e396703bfe1a1ae_lecture_06.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
18.917 Topics in Algebraic Topology: The Sullivan Conjecture
Fall 2007
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Steenrod Operations (Lecture 2)
The objective of today’s lecture is to introduce the Steenrod operations a... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/46c2e62c9fe4730096ed92d58d85783b_lecture2.pdf |
-module spectrum is the cochain complex
C ∗(X; F2)
of a topological space X. The cohomology groups of this F2-module spectrum are simply the cohomology
groups of X. The cohomology H∗(X; F2) has the structure of a graded commutative ring. The multiplication
on H∗(X; F2) arises from a multiplication which exists on t... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/46c2e62c9fe4730096ed92d58d85783b_lecture2.pdf |
we will denote by Dn(V ).
Remark 3. In concrete terms, Dn(V ) may be computed in the following way. Let M denote the vector
space F2, with the trivial action of Σn. Choose a resolution
. . .
→
P −1
→ →
P 0 M
by free F2[Σn]-modules. We let EΣn denote the complex P •. (We can think of EΣn as a contractible
comple... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/46c2e62c9fe4730096ed92d58d85783b_lecture2.pdf |
the consequences of the existence of a symmetric multiplication on a
complex V .
Notation 8. Let n be an integer. We let F2[−n] denote the complex which consists of a 1-dimensional
vector space in cohomological degree n, and zero elsewhere. Let en denote a generator for the F2-vector
space Hn F2[−n], so we have iso... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/46c2e62c9fe4730096ed92d58d85783b_lecture2.pdf |
homology
H (RP ∞; F2): this is just a one-dimensional vector space in each degree m, with a unique generator which
we will denote by xm.
∗
Definition 9. Let V be a complex, and let v ∈ Hn V , so that v determines a homotopy class of maps
For i ≤ n, we let
denote the image of
under the induced map
η : F2[−n] → V. ... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/46c2e62c9fe4730096ed92d58d85783b_lecture2.pdf |
9 yields operations
Sqi : Hn(X; F2)
→
Hn+i(X; F2).
These are the usual Steenrod operations.
i
v completely account for the cohomology groups of any extended
Remark 12. The operations v �→ Sq
square D2(V ). More precisely, let us suppose that V is an F2-module spectrum, and that {vi}i∈I is an
ordered basis for π∗V... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/46c2e62c9fe4730096ed92d58d85783b_lecture2.pdf |
to prove. If k = n, then
k
Sq
k
(v + v�) = (v + v�)2 = Sq
k
(v) + Sq
(v�) + (vv� + v�v).
Since the multiplication map
V ⊗ V → D2(V )
is commutative on the level of homotopy, we have vv� + v�v = 2vv� = 0.
Now suppose that k < n. By functoriality, it will suffice to treat the universal case where V � F[−n] ⊕
F[−n]. Us... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/46c2e62c9fe4730096ed92d58d85783b_lecture2.pdf |
8.022 (E&M) – Lecture 9
Topics:
RC circuits
Thevenin’s theorem
Last time
Electromotive force:
How does a battery work and its internal resistance
How to solve simple circuits:
Kirchhoff’s first rule: at any node, sum of the currents in = sum
of the currents out (conservation of charge at node... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/46de97e4978fc003565f683871741451_lecture9.pdf |
at any moment in time
tage drop on the resistor: -IR
Vol
Æ Q(t) Æ
V(t)
Q
C
−
IR
0
=
Not useful in this form since I=I(Q)
I=-dQ/dt (- sign because C is losing charge)
Q dQ
+
dt
C
0R
=
Easy integral yields to exponential decay of the charge:
−
( )
Q t Q e
=
0
t
RC
G. Sciolla – MIT
8.022 –... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/46de97e4978fc003565f683871741451_lecture9.pdf |
t
RC
d
dt
−
e
⎛
⎜
⎝
⎞
⎟
⎠
=
Q
0
RC
t
RC
−
e
Same exponential decay as for Q(t)
G. Sciolla – MIT
8.022 – Lecture 9
7
Charging capacitors
Now 3 elements in circuit: EMF, capacitor and resistor
Capacitor starts uncharged
C
C
- - - - - - -
+ + + + + + +
s
s
I
V
+ -
R
R
What happens wh... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/46de97e4978fc003565f683871741451_lecture9.pdf |
(
−
RC
)
'
dQ
Q
'
Integrating between t=0 and t:
⇒
= −
dt
RC
Q Q t
=
( )
∫
Q
=
0
t
dQ
'
∫
= −
Q
'
t
t
=
=
0
dt
RC
⇒
ln
Q t CV
( ) -
CV
-
= −
t
RC
⇒
Q
CV
( ) -
t
V
C
−
t
RC
=
e
−
Q
t
( )
=
C
t
RC
−
e
V
⎛
1
−⎜
⎝
⎞
⎟
⎠
G. Sciolla – MIT
8.022 – Lecture 9
10
5
... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/46de97e4978fc003565f683871741451_lecture9.pdf |
I
( )
t
=
t
R C
−
e
V
R
Are Kirchhoff’s laws valid at any moment in time?
V
−
Q
C
−
IR
V
= −
V
t
R C
−
e
⎛
1
⎜
⎝
−
⎞
⎟
⎠
−
R
t
R C
−
e
V
R
=
0
O K !
Asymptotic behavior of the capacitor:
At t=0: I=V/R as if C were a short circuit
At t=infinity
, I=0 as if C were an open circu
it... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/46de97e4978fc003565f683871741451_lecture9.pdf |
9
13
Verify time constant (E8)
RC circuit with
VEMF = squared 5 V pulses
Variable C initially = 0.3
µF
Variable R initially = 400
Ω
2
R1 = 100 Ω
Display on scope V
C and I(R1)
Verify that time constant is RC
VC(t)
5V
1-e-t/RC
IAG(t)
10mA
-t/RC
e
R2
V
EMF
C
G
R1
A
G. Sciolla – ... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/46de97e4978fc003565f683871741451_lecture9.pdf |
8.022 – Lecture 9
16
8
Thevenin equivalence
Thevenin’s theorem:
Any combination of resistors and EMFs with 2 terminals can be replaced with a
series of a battery V
and a resistor R
OC
is the open circuit voltage
T where
VOC
RT=VOC/I
or RT=Req wi
th all the EMF shorted
short where I
short
is the curren... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/46de97e4978fc003565f683871741451_lecture9.pdf |
There is on y one current go ng through the reduced c rcuit
At t=0, C behaves like a short
At t=0 I
short=VOC/RT
Æ
i
l
with I
i
Æ RT=VOC/I
short
G. Sciolla – MIT
8.022 – Lecture 9
18
9
Solve the actual problem
Calculate VOC and RT=VOC/I
short
for our problem:
R1
V
+
-
R2
+
-
C
VOC
≡
+... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/46de97e4978fc003565f683871741451_lecture9.pdf |
:
VOC≡
+
-
RT
Any
unknown
combination
of Rs and EMFs
Careful:
Thevenin works only when the elements in the box follow Ohm’s law,
i.e.
on between V and I
linear relati
G. Sciolla – MIT
8.022 – Lecture 9
20
10
Oscillating circuit (E13)
RC circuit with:
VEMF = 1 kV
C = 0.1 µF
R = 2.5 M... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/46de97e4978fc003565f683871741451_lecture9.pdf |
022 – Lecture 9
22
11
Norton’s theorem
Any combination of resistors and EMFs with 2 terminals can be replaced with a
parallel of a current generator IN
RT
and a resistor R
is the equivalent resistance of the circuit w th all the EMF shorted and all the
T where
i
current sources open (same as Thevenin!)
IN ... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/46de97e4978fc003565f683871741451_lecture9.pdf |
Lecture 8
Primitive Roots (Prime Powers), Index Calculus
Recap - if prime p, then there’s a primitive root g mod p and it’s order mod p
is p − 1 = qe1 e2
1 q2 . . . qr . We showed that there are integers gi mod p with order
i − 1 ≡ 0 mod p). Set g = (cid:81) gi -
qei
(counting number of solutions to xq i
exactly
i
(cid... | https://ocw.mit.edu/courses/18-781-theory-of-numbers-spring-2012/4725535fce239230067825c86e89afaa_MIT18_781S12_lec8.pdf |
Then there are infinitely many primes p for which a is a primite root mod p.
This is an open question. Hooley proved this conditional on GRH, and Heath-
Brown showed that if a is a prime, then there are at most 2 values of a which
fail the conjecture
(Definition) Discrete Log: Say p is a prime, and g is a primitive root ... | https://ocw.mit.edu/courses/18-781-theory-of-numbers-spring-2012/4725535fce239230067825c86e89afaa_MIT18_781S12_lec8.pdf |
0 mod p),
we can write a ≡ gl mod p so if x ≡ gk as before then gkd ≡ gl mod p. This
means that gkd−l ≡ 1 mod p ↔ p − 1|kd − l ↔ kd ≡ l mod p − 1 (k is variable),
which has a solution iff (d, p − 1) divides l, in which case it has exactly (d, p − 1)
solutions.
Note:
(d, p − 1) divides l ←→ p − 1 divides
l(p
− 1)
(d, p ... | https://ocw.mit.edu/courses/18-781-theory-of-numbers-spring-2012/4725535fce239230067825c86e89afaa_MIT18_781S12_lec8.pdf |
.
2 Since there are p possible values of t(0 ≤ t ≤ p − 1), any of these remaining ones
give a g + tp which is a primitive root mod p2. Consider f (x) = xp−1
1: mod
− and f (cid:48)(g) = (p − 1)gp−2 (cid:54)≡ 0 mod p,
p it has the root g. Since f (cid:48)(x) = (p − 1)xp 2
by Hensel’s Lemma there is a unique lift g + tp ... | https://ocw.mit.edu/courses/18-781-theory-of-numbers-spring-2012/4725535fce239230067825c86e89afaa_MIT18_781S12_lec8.pdf |
−1
≡
1 mod pe−1 assuming that
) = 1 + bpe−1 with p (cid:45) b. Need to
We know that gpe−2(p−1) = 1 + bpe−1. Raising to power p we get
gpe−1(p−1) = (1 + bpe−1)p
= 1 + pbpe−1
+
(bpe−1
)2 +
(cid:18) (cid:19)
p
3
(bpe−1)3 + . . .
(cid:18)
(cid:19)
p
2
pe
≡ 1 + bpe mod
+1
(cid:0) (cid:1)
(cid:0) (cid:1)
2 so p b2p2e
(becaus... | https://ocw.mit.edu/courses/18-781-theory-of-numbers-spring-2012/4725535fce239230067825c86e89afaa_MIT18_781S12_lec8.pdf |
1 mod m and aφ(m(cid:48)) ≡ 1 mod m(cid:48). So
aφ(m)φ(m(cid:48))/2 ≡ (aφ(m))φ(m(cid:48))/2
≡ 1 mod m
aφ(m)φ(m(cid:48))/2 ≡ 1 mod m(cid:48)
Similarly so, aφ(m)φ(m(cid:48))/2 ≡ 1 mod n
but φ(n) = φ(m)φ(m(cid:48)) so ordn(a) < φ(n). So a can’t be a primitive root mod n.
Only remaining candidate is n = 2k for k ≥ 3. No pr... | https://ocw.mit.edu/courses/18-781-theory-of-numbers-spring-2012/4725535fce239230067825c86e89afaa_MIT18_781S12_lec8.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.189 Multicore Programming Primer, January (IAP) 2007
Please use the following citation format:
Saman Amarasinghe, 6.189 Multicore Programming Primer, January
(IAP) 2007. (Massachusetts Institute of Technology: MIT
OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). ... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
ism
●
Interleaved Concurrency
Logically simultaneous processing B
Interleaved execution on a single
C
processor
A
● Parallelism
Physically simultaneous processing
Requires a multiprocessors or a
multicore system
A
B
C
Time
Time
Prof. Saman Amarasinghe, MIT.
4
6.189 IAP 2007 MIT
Account and Ban... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
String pass = in.readLine();
if (!acc.is_password(pass))
throw new Exception();
out.print(“yourbalance is“ + acc.getbal());
out.print("Depositor withdraw amount > “ );
int val = in.read();
if (acc.getbal() + val > 0)
acc.post(val);
else
throw new Exception();
out.print(“yourbalance is“ + acc.getbal());
} cat... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
{
out.println("Invalidinput, restart“ );
I need to run multiple ATM machines from my program, how do I do that?
8
Prof. Saman Amarasinghe, MIT.
6.189 IAP 2007 MIT
}
}
}
}
Concurrency in Java
● Java has a predefined class java.lang.Thread which provides
the mechanism by which threads are created
public class ... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
();
out.print("Password>“ );
String pass = in.readLine();
if (!acc.is_password(pass))
throw new Exception();
out.print(“yourbalance is“ + acc.getbal());
out.print("Depositor withdraw amount > “ );
int val = in.read();
if (acc.getbal() + val > 0)
acc.post(val);
else
throw new Exception();
out.print(“yourbala... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
} catch(Exception e) {
out.println("Invalidinput, restart“ );
}
}
I need to run multiple ATM machines from my program, how do I do that?
12
Prof. Saman Amarasinghe, MIT.
6.189 IAP 2007 MIT
Activity trace
ATM 1
Account ID >
allyssa
Password >
MITROCKS
ATM 2
Account ID >
ben
Password >
i
T
m
e
Your account balanc... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
acc.getbal());
Your account balance is 10
out.print(“yourbalance is“ + acc.getbal());
Your account balance is 10
Prof. Saman Amarasinghe, MIT.
15
6.189 IAP 2007 MIT
Activity trace II
balanc ATM 1
e
100
void post(int v) {
100
100
10
10
v
}
ATM 2
void post(int v) {
balance = balance +
v
-90
balance =
+ v
... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
6.189 IAP 2007 MIT
Why You Need Locks
thread A
thread B
if (no milk && no note)
if (no milk && no note)
leave note
buy milk
remove note
leave note
buy milk
remove note
Milk
Vitamin
D
Milk
Vitamin
D
Image by MIT OpenCourseWare.
Image by MIT OpenCourseWare.
● Does this work?too much milk
Prof. Saman Amarasinghe,... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
code can be labeled as synchronized
● The synchronized keyword takes as a parameter an object
whose lock the system needs to obtain before it can
continue
● Example:
synchronized (acc) {
if (acc.getbal() + val > 0)
acc.post(val);
else
throw new Exception();
out.print(“yourbalance is “ + acc.getbal());
}
Prof... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
IAP 2007 MIT
Activity trace II
balance
ATM 1
100
100
100
100
10
10
10
out.print(“yourbalance is“ + acc.getbal());
Your account balance is 100
out.print("Depositor withdraw amount >“);
Deposit or Withdraw amount >
-90
int val = in.read();
synchronized(acc)
if (acc.getbal() + val > 0)
acc.post(val);
o... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
bnk.get(id);
if (acc == null) throw new Exception();
out.print("Password>“ );
String pass = in.readLine();
if (!acc.is_password(pass))
throw new Exception();
synchronized (acc) {
out.print(“yourbalance is “ + acc.getbal());
out.print("Depositor withdraw amount >“ );
int val = in.read();
if (acc.getbal() + val > ... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
() > val)
from.post(-val);
synchronized(to)
Waiting for Allyssa’s account
to be released to perform
DEADLOCKED!
Prof. Saman Amarasinghe, MIT.
29
6.189 IAP 2007 MIT
Avoiding Deadlock
● Cycle in locking graph = deadlock
● Standard solution:
canonical order for locks
Acquire in increasing order
Release in... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
2007 MIT
Races
Race conditions – insidious bugs
Non-deterministic, timing dependent
Cause data corruption, crashes
Difficult to detect, reproduce, eliminate
● Many programs contain races
Inadvertent programming errors
Failure to observe locking discipline
Prof. Saman Amarasinghe, MIT.
33
6.189 IAP... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
philosopher thinks for a while.
When the philosopher becomes hungry, she
stops thinking and…
– Picks up left and right chopstick
– He cannot eat until he has both chopsticks, has to
wait until both chopsticks are available
– When the philosopher gets the two chopsticks she
eats
When the philosopher is done ... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
;
public void run() {
try {
while(true) {
1
2
3
4
synchronized(table) {
synchronized(left) {
synchronized(right) {
System.out.println(times + ": Philosopher " + position + " is done eating");
}
}
}
}
} catch (Exception e) {
System.out.println("Philosopher " + position + "'s meal got disturbed");
}
} ... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
.
44
6.189 IAP 2007 MIT
Conclusion
● Concurrency and Parallelism are important concepts
in Computer Science
● Concurrency can simplify programming
However it can be very hard to understand and debug
concurrent programs
● Parallelism is critical for high performance
From Supercomputers in national labs to ... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
3.044 MATERIALS PROCESSING
LECTURE 9
Example 1: Casting into low conductivity molds
Greatly simplified if:
1. Mold is thick → can neglect air → semi-infinite geometry → erf
2. Mold controls heat loss → Tmold increases → kmold decreases →
3. Superheating is negligible → liquid poured at Tm
gradients are
in mold
D... | https://ocw.mit.edu/courses/3-044-materials-processing-spring-2013/47576d530e3a052c0c3b2451f4076eb6_MIT3_044S13_Lec09.pdf |
)
k
mρmcp,m
t
(cid:3) 1
2
s =
2(T0
− Tm)
ρsHf
Boundary Condition: @t = 0, s = 0
Boundary Condition: @s = L, t = tf
3.044 MATERIALS PROCESSING
3
tf ∝ L2
V
A
(cid:2)
L ≈
tf ∝
V
A
(cid:3)
2
⇒ Chvorinov’s Rule
Example 2: Thin castings / Cold molds / Highly conductive molds
out
(cid:9)
(cid:6)
(cid:7)(cid:8)
−h(Tm − Tmold
... | https://ocw.mit.edu/courses/3-044-materials-processing-spring-2013/47576d530e3a052c0c3b2451f4076eb6_MIT3_044S13_Lec09.pdf |
Machine Learning for Healthcare
HST.956, 6.S897
Lecture 5: Risk stratification (continued)
David Sontag
1
Outline for today’s class
1. Risk stratification (continued)
– Deriving labels
– Evaluation
– Subtleties with ML-based risk stratification
2. Survival modeling
2
Where ... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/476c3cdeaf64631d2fb0332832ff6250_MIT6_S897S19_lec5.pdf |
AUC =
Probability that
algorithm ranks
a positive
patient over a
negative patient
Invariant to
amount of class
imbalance
Diabetes
1-year gap
False positive rate
8
3-.-4?-1H9F-17/91).N717./-145/4.).01?-
!"#$%
#&'(&")"*(&"+,
0507LLD)89.05-5)
9:)c05/... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/476c3cdeaf64631d2fb0332832ff6250_MIT6_S897S19_lec5.pdf |
-fair-use/
12
Non-stationarity:
Top 100 lab measurements over time
s
b
a
L
Time (in months, from 1/2005 up to 1/2014)
→ Significance of features may change over time
© Narges Razavian. All rights reserved. This content is excluded from our Creative Commons lic... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/476c3cdeaf64631d2fb0332832ff6250_MIT6_S897S19_lec5.pdf |
Caruana et al., Intelligible Models for Healthcare: Predicting Pneumonia Risk and Hospital 30-
day Readmission. KDD 2015.]
16
Intervention-tainted outcomes
• Formally, this is what’s happening:
ED triage
Tr... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/476c3cdeaf64631d2fb0332832ff6250_MIT6_S897S19_lec5.pdf |
No big wins from deep models on
structured data/text
Supplemental Table 1: Prediction accuracy of each task of deep learning model compared to baselines
Inpatient Mortality,
AUROC1
(95% CI)
Deep learning 24 hours after admission
Full feature enhanced baseline at 24 hours after admission
Full feature simple baseline ... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/476c3cdeaf64631d2fb0332832ff6250_MIT6_S897S19_lec5.pdf |
.86)
0.83 (0.83-0.84)
0.81 (0.80-0.82)
0.74 (0.73-0.75)
Courtesy of Rajkomar et al. Used under CC BY.
21
No big wins from deep models on
structured data/text
Comparison
to Razavian
et al. ‘15
Supplemental Table 1: Prediction accuracy of each task of deep learning model compared to baselines
In... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/476c3cdeaf64631d2fb0332832ff6250_MIT6_S897S19_lec5.pdf |
7 days AUROC (95% CI)
Deep learning 24 hours after admission
Full feature enhanced baseline at 24 hours after admission
Full feature simple baseline at 24 hours after admission
Baseline (mLiu4) at 24 hours after admission
0.86((0.86
-0.87
0.85 (0.84-0.85)
0.83 (0.82-0.84
(
((0.75
-0.77
0.76
-0.86
(0.85
0.85(
0.83 (
(0.... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/476c3cdeaf64631d2fb0332832ff6250_MIT6_S897S19_lec5.pdf |
reserved. This content is excluded from our Creative Commons license. For more information, see https://ocw.mit.edu/help/faq-fair-use/
25
Survival modeling
• Why not use classification, as before?
– Less data for training (due to exclusions)
– Pessimistic estimates due to choice of... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/476c3cdeaf64631d2fb0332832ff6250_MIT6_S897S19_lec5.pdf |
12
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 1, JANUARY 2004
A Laguerre Polynomial-Based Bound on the Symbol Error Probability for
Adaptive Antennas With Optimum Combining
Marco Chiani, Senior Member, IEEE, Moe Z. Win, Fellow, IEEE, Alberto Zanella, Member, IEEE, and
Jack H. Winters, Fellow, IEEE
... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
ing (PSK) have been
derived for the case of a single nonfading interferer with Rayleigh
fading of the desired signal in [1] and [2] and with Rayleigh fading
of the desired signal and a single interferer in [3]. In [4], two dif
ferent methods (direct and moment generation function-based ap
proaches), requiring a si... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
Digital Object Identifier 10.1109/TWC.2003.821165
multiple interferers of arbitrary powers, closed-form expressions
of the BEP for PSK with OC are not available in the literature:
Thus, Monte Carlo simulation has been used to determine the BEP
in [2]. Unfortunately, such simulations are computation intensive
and n... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
ibel), and therefore, in general,
significantly better than previous bounds for the case of multiple
equal-power interferers.
In Section II, we describe the system model. Performance and
upper bounds are derived in Section III, and in Section IV we
compare our analytical bounds with Monte Carlo simulations.
II. S... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
can be written as
where
denoted by
values with
as those of the
are the eigenvalues of a complex Wishart matrix,2
eigen
can be thought of
[9]. Hence, the first
of
matrix
(1)
(6)
and
and
and
are the mean (over fading) energies of the
where
desired and interfering signals, respectively;
are the
desired... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
ers and thermal noise in a fading environment is now ob
tained by averaging the conditional SEP over the (desired and
,
interfering signals) channel ensemble as
is the SEP conditioned on the random variable
where
. This can be accomplished by using the chain rule of condi
tional expectation as
(7)
where we fir... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
where
together with the fact that
conditional SEP
antennas and
, conditioned on
interferers, becomes
. Using (8)
and
is Gaussian with i.i.d. elements, the
, in the general case of
Then, by remembering that
, and the fact that
’s are real and nonnegative, the following inequality
and
holds:
Therefore, by ... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
, eq. (39)], we see that (14) is the
-ary
exact expression of the SEP for coherent detection of
-branch MRC in the absence of interference. Note
PSK using
that
in (13) is independent of interference, and de
pends only upon the SNR and the number of antenna elements.
Other factors in (13) are independent of SNR,... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
20 22 24 26 28 30
SNR [dB]
Fig. 2.
Comparison of the bounds on SEP for coherent detection of binary
PSK using OC with four antennas and SIR � �� dB in the presence of one
and four interferers. Also shown in the figure are the results obtained by Monte
Carlo simulations.
is a monic polynomial of degree
with nonn... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
known bound, e.g.,
the difference is more than 4 dB at SEP
.
Fig. 3 shows the SEP as a function of SIR for several values
of SNR, for
, and binary phase-shift keying
(BPSK) modulation. The comparison with simulation results
shows that, for a reasonable range of SIR values, the proposed
upper bound is tight for ... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
to three, an 8-PSK modulation scheme is considered with
dB. Performance with MRC is evaluated by using
SIR
is given
, where
by (14). Also shown in the figure are Monte Carlo simulation
results for OC. The results show that, as expected, for small
SNR, the thermal noise is dominant and, therefore, MRC and
OC per... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
SIR)-plane has been obtained for
. Note that the
8 PSK. The curves are for
two asymptotes (vertical and horizontal) give the values of SIR
and SNR without thermal noise and interference, respectively.
The region below each curve represents the outage domain re
gion in which all points produce an SEP higher than t... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
,” Telecommun. Radio Eng., vol. 34/35, pp.
83–85, Oct. 1980.
[2] J. H. Winters, “Optimum combining in digital mobile radio with
cochannel interference,” IEEE J. Select. Areas Commun., vol. SAC-2,
pp. 528–539, July 1984.
[3] V. A. Aalo and J. Zhang, “Performance of antenna array systems with
optimum combining in a... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
multiple antennas,” IEEE Wireless
Pers. Commun., vol. 6, pp. 311–335, Mar. 1998.
[9] M. Chiani, M. Z. Win, A. Zanella, and J. H. Winters, “Exact symbol
error probability for optimum combining in the presence of multiple
co-channel interferers and thermal noise,” in Proc. IEEE Global
Telecommunications Conf., vol. ... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
M. Z. Win and J. H. Winters, “Virtual branch analysis of symbol error
probability for hybrid selection/maximal-ratio combining in Rayleigh
fading,” IEEE Trans. Commun., vol. 49, pp. 1926–1934, Nov. 2001.
[16] M. K. Simon, S. M. Hinedi, and W. C. Lindsey, Digital Communication
Techniques: Signal Design and Detection... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
6.720J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 9-1
Lecture 9 - Carrier Flow (cont.)
February 23, 2007
Contents:
1. Shockley’s Equations
2. Simplifications of Shockley equations to 1D quasi-neutral
situations
3. Majority-carrier type situations
Reading assignment:
del Alamo, Ch. 5, §§5.3-5... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/477625fc6080d062ea9bef7e100acd0b_lecture9.pdf |
2)ve
drift
+ qDe∇n
(cid:2)
−)
+ − NA
Hole current equation:
(cid:2)
Jh = qp (cid:2)vh
drift
− qDh∇p
(cid:2)
Electron continuity equation:
∇ · J(cid:2)
(cid:2)
∂t = Gext − U (n, p) + 1
∂n
e
q
Hole continuity equation:
∇ · J(cid:2)
(cid:2)
∂p
∂t = Gext − U (n, p) − 1
h
q
Total current equation:
t = J(... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/477625fc6080d062ea9bef7e100acd0b_lecture9.pdf |
of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 9-5
Shockley’s equations in 1D:
Gauss’ law:
E = q
∂
∂x
(cid:2)(p − n + ND − NA)
Electron current equation:
Je = −qnvdrift (E) + qDe
e
∂n
∂x
Hole current equation:
drift (E) − qDh
Jh = ... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/477625fc6080d062ea9bef7e100acd0b_lecture9.pdf |
situation
(field independent
of n, p)
Majority-carrier
type situation
(V=0, n'=p'=0)
Minority-carrier
type situation
(V=0, n'=p'=0, LLI)
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of T... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/477625fc6080d062ea9bef7e100acd0b_lecture9.pdf |
N
D
−−N
NA
+
D
−− NA
| (cid:6) 1
which implies
n
o
− (cid:4)
p
o
+
−N
D
N
−
A
• Additionally, quasi-neutrality outside equilibrium:
|
p(cid:5) − n(cid:5)
n
(cid:5)
| (cid:4) |
p(cid:5) − n(cid:5)
p
(cid:5)
| (cid:6) 1
which implies:
p (cid:5) (cid:4) n (cid:5)
• QN approximation good if n, p high ⇒ carriers ... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/477625fc6080d062ea9bef7e100acd0b_lecture9.pdf |
://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 9-10
2. Subtract one continuity equation from the other:
∂Jt
∂x
= q
∂(n − p)
∂t
= −
∂ρ
∂t
continuity equation for net volume charge: if Jt changes with... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/477625fc6080d062ea9bef7e100acd0b_lecture9.pdf |
Jt (cid:4) 0
∂x
Jt = Je + Jh
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 9... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/477625fc6080d062ea9bef7e100acd0b_lecture9.pdf |
carrier current (n-type):
Must distinguish between internal field in TE (Eo) and total field
outside equilibrium (E).
For simplicity, do in low-field limit (exact case done in notes).
In equilibrium:
Jeo = qμenoEo + qDe
dno
dx
= 0
Out of equilibrium:
Hence:
dno
Je (cid:4) qμenoE + qDe dx
Je = qμeno(E − Eo) = qμenoE... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/477625fc6080d062ea9bef7e100acd0b_lecture9.pdf |
� Example 1: Integrated Resistor with uniform doping (n-type)
z
y
L
W
n
p
n+
n+
top view
x
x
n+
t
n
n+
cross-sectional view
p
Uniform doping ⇒ Eo = 0, then:
Jt = −qNDve
drift (E)
• If E not too high,
I-V characteristics:
Jt (cid:4) qNDμeE
V
I = W tqNDμe L
Cite as: Jesús del Alamo, course materials ... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/477625fc6080d062ea9bef7e100acd0b_lecture9.pdf |
8)= Jt(x)
• Majority carrier-type situations characterized by application
of external voltage without perturbing carrier concentrations.
• Majority-carrier type situations dominated by drift of majority
carriers.
• Integrated resistor:
– for low voltages, current proportional to voltage across
– for high voltage... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/477625fc6080d062ea9bef7e100acd0b_lecture9.pdf |
Introduction to Algorithms: 6.006
Massachusetts Institute of Technology
Instructors: Erik Demaine, Jason Ku, and Justin Solomon
Lecture 1: Introduction
Lecture 1: Introduction
The goal of this class is to teach you to solve computation problems, and to communicate that
your solutions are correct and efficient.
Pr... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-spring-2020/477c78e0af2df61fa205bcc6cb613ceb_MIT6_006S20_lec1.pdf |
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