text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
Kamerlingh Onnes
the Netherlands
Leiden University
Leiden, the Netherlands
b. 1853
d. 1926
•http://www.nobel.se/physics/laureates
5
Discovery of Superconductivity
“As has been said, the experiment left
no doubt that, as far as accuracy of
measurement went, the resistance
disappeared. At the same time,
however, s... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/41b2882232b1b43823ec488969c21b6d_lecture1.pdf |
for copyright reasons.
Please see: "A current-carrying type II
superconductor in the mixed state" from
http://phys.kent.edu/pages/cep.htm
http://phys.kent.edu/pages cep.htm
/
)
)
When a current is applied to a type II superconductor (blue
rectangular box in the m xed state, the magnetic vortices
(blue cylinder... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/41b2882232b1b43823ec488969c21b6d_lecture1.pdf |
interplay of theory and experiment.
It would have been very difficult to have arrived at the
theory by purely deductive reasoning from the basic
equations of quantum mechanics. Even if someone had
done so, no one would have believed that such
remarkable properties would really occur in nature.
But, as you well kn... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/41b2882232b1b43823ec488969c21b6d_lecture1.pdf |
for copyright reasons.
Please see: http://nobelprize.org/physics/laureates/1973/index.html
Leo Esaki
Ivar Giaever Brian David
Josephson
1/4 of the prize
1/4 of the prize
1/2 of the prize
Japan
USA
United Kingdom
IBM Thomas J.
Watson Research
Center
Yorktown Heights,
NY, USA
b. 1925
General Electric
Co... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/41b2882232b1b43823ec488969c21b6d_lecture1.pdf |
) ⋅ l d
dI
V = LJ dt
,
where LJ =
Φ 0
2π I cosϕc
Φ 0 =
flux
quantum
483.6
GHz
/ mV
19
20
•10
SQUID Magnetometers
Φ0
I+
V+
1pF
Φ
1.1
µm
1.1
µm
I
V
20 µm
DC SQUID
Shunt capacitors ~ 1pF
Jct. Size ~ 1.1µm
Loop size ~20x20µm2
L
SQUID ~ 50pH
I ~10 & 20µA
c
21
High-Temperature Superconductivity
... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/41b2882232b1b43823ec488969c21b6d_lecture1.pdf |
________
laureates/1987/bednorz-lecture.html
http://www.nobel.se/physics/laureates/1987/bednorz-muller-lecture.pdf
23
Perovskite Structure
Image removed for copyright reasons.
______________________________________
Please see: Images from http://cst-www.nrl.navy.mil/lattice/struk/perovskite.html
•http://cst-www.n... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/41b2882232b1b43823ec488969c21b6d_lecture1.pdf |
MRS meetings since
their discovery in 1986. Twenty years later, the progress both on the fundamental
understanding of these materials and the path towards their industrial applications has
been impressive. First-generation wires are now routinely produced in kilometer
lengths and used in a variety of large-scale pr... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/41b2882232b1b43823ec488969c21b6d_lecture1.pdf |
Microwave filters in cellular stations
Technical Points
Low losses, smaller size, sharp filtering
Passive microwave devices,
Resonators for oscillators
Far-infrared bolometers
Microwave detectors
X-ray detectors
SQUID Magnetometers:
Magneto-encephalography, NDT
Lower surface losses, high quality
factors, sma... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/41b2882232b1b43823ec488969c21b6d_lecture1.pdf |
in a
superconducting loop
•
The induced current can
be in the opposite
direction if we carefully
choose a different
magnetic field this time
Image removed for copyright reasons.
•
To store and process information
as a computer bit, we assign:
as state
| 0 〉
as state
| 1 〉
clockwise
Anti-clockwise
35
... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/41b2882232b1b43823ec488969c21b6d_lecture1.pdf |
Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS
by A. Megretski
Lecture 3: Continuous Dependence On Parameters1
Arguments based on continuity of functions are common in dynamical system analysis.
They rarely apply to... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/41d1588bb21baf8002a8b9b56ed1047c_lec3_6243_2003.pdf |
require existence
of a constant M such that
2
|a(¯
x1) − a(¯
x2)| ∃ M |x1 − ¯
x2|
¯
x1, ¯
[t0, tf ] ∈� Rn of (3.1). The proof of both
for all ¯ x2 from a neigborhood of a solution x :
existence and uniqueness is so simple in this case that we will formulate the statement
for a much more general class of integral e... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/41d1588bb21baf8002a8b9b56ed1047c_lec3_6243_2003.pdf |
(x(� ), �, t)d� � t ≤ [t0, tf ].
(3.4)
t0
A proof of the theorem is given in the next section. When a does not depend on the
third argument, we have the standard ODE case
x˙ (t) = a(x(t), t).
In general, Theorem 3.1 covers a variety of nonlinear systems with an infinite dimensional
state space, such as feedback i... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/41d1588bb21baf8002a8b9b56ed1047c_lec3_6243_2003.pdf |
t0 ∃ 1/(2K) we have
|xk+1(t) − xk (t)| ∃
t
t
t0
|a(xk (� ), �, t) − a(xk−1(� ), �, t)|d�
∃ K|xk (� ) − xk−1(� )|d�
t0
∃ 0.5 max {|xk (t) − xk−1(t)|}.
t�[t0,tf ]
Therefore one can conclude that
max {|xk+1(t) − xk (t)|} ∃ 0.5 max {|xk (t) − xk−1(t)|}.
t�[t0,tf ]
t�[t0,tf ]
Hence xk (t) converges exponenti... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/41d1588bb21baf8002a8b9b56ed1047c_lec3_6243_2003.pdf |
the same proof applies when (3.2),(3.3) are
replaced by the weaker conditions
|a(¯
x1, �, t) − a(¯
x2, �, t)| ∃ K(� )|¯
x1 − ¯
x2| � ¯ x2 ≤ Br (¯
x1, ¯
x0), t0 ∃ � ∃ t ∃ t1,
and
x, �, t)| ∃ m(t) � ¯
where the functions K(·) and M (·) are integrable over [t0, t1].
x ≤ Br (¯
x0), t0 ∃ � ∃ t ∃ t1,
|a(¯
4
3.2 Continuo... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/41d1588bb21baf8002a8b9b56ed1047c_lec3_6243_2003.pdf |
R such that
|a(¯
x1, �, t, q)−a(¯
x2, �, t, q)| ∃ K|x1−x2| � ¯ x2 ≤ X d, t0 ∃ � ∃ t ∃ tf , q ≤ (q0−d, q0+d);
x1, ¯
¯
¯
(b) there exists K ≤ R such that
|a(¯
x, �, t, q)| ∃ M � ¯
x ≤ X d, t0 ∃ � ∃ t ∃ tf , q ≤ (q0 − d, q0 + d);
(c) for every � > 0 there exists � > 0 such that
|x0(q1) − ¯
¯
x0(q2)| ∃ � � q1, q2 ≤ (q... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/41d1588bb21baf8002a8b9b56ed1047c_lec3_6243_2003.pdf |
x0 . Condition
(b) simply bounds a uniformly. Finally, condition (c) means continuous dependence of
equations and initial conditions on parameter q.
The proof of Theorem 3.2 is similar to that of Theorem 3.1.
3.3
Implications of continuous dependence on parameters
This section contains some examples showing how ... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/41d1588bb21baf8002a8b9b56ed1047c_lec3_6243_2003.pdf |
to x(t) = x(t − t0, ¯
x) means “the value x(t) of the
solution of (3.8) with initial conditions x(0) = ¯x”. Remember that this definition makes
sense only when uniqueness of solutions is guaranteed, and that x(t, ¯x) may by undefined
when |t| is large, in which case we will write x(t, ¯x) = ⊂.
x0), where x(t, ¯
Accor... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/41d1588bb21baf8002a8b9b56ed1047c_lec3_6243_2003.pdf |
, i.e. x(t, ¯
x0) ≥ ¯
x0 is a
(a) there exists d > 0 such that x(t, ¯
x) � ¯
x0 as t � ⊂ for all ¯
x satisfying |¯
x0 − ¯
x| < d;
(b) for every � > 0 there exists � > 0 such that |x(t, ¯
x) − ¯
x0| < � whenever t → 0 and
x − ¯
|¯
x0| < �.
In other words, all solutions starting sufficiently close to an asymptotically s... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/41d1588bb21baf8002a8b9b56ed1047c_lec3_6243_2003.pdf |
converges to infinity, is called the limit set of the “trajectory”
t ∈� x(t, ¯x0).
x0) � x
Theorem 3.4 The limit set of a given trajectory is always closed and invariant under
the transformations ¯
x ∈� x(t, ¯
x). | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/41d1588bb21baf8002a8b9b56ed1047c_lec3_6243_2003.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
18.969 Topics in Geometry: Mirror Symmetry
Spring 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
MIRROR SYMMETRY: LECTURE 4
DENIS AUROUX
1. Pseudoholomorphic Curves
For (X 2n, ω) symplectic, J a compatible a.c.s. ∈ J ... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/41f318431154e73d835ed92bc5c40ab4_MIT18_969s09_lec04.pdf |
− 6 + 2k)
u is regular if D∂ is onto.
Theorem 1. The set J reg(X, β) of J ∈ J (X, ω) s.t. every simple J-holomorphic
curve in class β is regular is a Baire subset. For J ∈ J reg(X, β), the subset of
simple maps M∗ (X, J, β) ⊂ Mg,k(X, J, β) is smooth and oriented of dimension
2d.
g,k
Let g(·, ) = ω( , J ) be the a... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/41f318431154e73d835ed92bc5c40ab4_MIT18_969s09_lec04.pdf |
, ∞ to CP1 × CP1 . Away from
x = 0, it converges uniformly to x �→ (x, 0). But if you reparameterize to ˜x = nx,
1 ) and away from x = ∞, it converges uniformly to ˜ → (0, 1
x ).
x˜ �→ ( n
˜
1 ˜ x
x, ˜
x
nx
The general idea is:
• Identify bubbling regions where sup |dun| → ∞.
• Away from those, ∃ convergent s... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/41f318431154e73d835ed92bc5c40ab4_MIT18_969s09_lec04.pdf |
�→ u(zi) for 1 ≤ i ≤ k. The Gromov-Witten invariants
are defined as follows: given α1, . . . , αk ∈ H ∗(X),
deg (αi) = 2d,
�
(8)
�α1, . . . , αk�g,β =
ev∗
1α1 ∧ · · · ∧ evk
∗αk ∈ Q
�
[M g,k(X,J,β)]
i
ev−1(Ci)) (or rather #(ev
X (choose Ci trans
Equivalently, if we represent P D(αi) by a cycle Ci
verse to th... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/41f318431154e73d835ed92bc5c40ab4_MIT18_969s09_lec04.pdf |
,
M0,k(X, J, dβ) are orbifolds (strata of multiply-covered maps are orbifolded). We
can restore transversality by taking domain-dependent Js. More precisely, there
d 1
β
MIRROR SYMMETRY: LECTURE 4
3
is a universal curve C → M0,k (the fiber over a point is the corresponding curve),
and J is now given by a map C ... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/41f318431154e73d835ed92bc5c40ab4_MIT18_969s09_lec04.pdf |
cover.
∞ →
For an algebraic geometer, one needs to keep J integrable so X remains an al
gebraic variety. The moduli space Mg,k is an algebraic stack, as is Mg,k(X, J, β).
For an integrable J and fixed j, we have a ∂-operator on sections of u∗T X, and
the cokernel of this operator is precisely H 1(Σ, u∗T X). Where du... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/41f318431154e73d835ed92bc5c40ab4_MIT18_969s09_lec04.pdf |
MIT 6.035
Semantic Analysis
Martin Rinard
Laboratory for Computer Science
Massachusetts Institute of Technology
Error Issue
• Have assumed no problems in building IR
• But are many static checks that need to be done
as part of translation
• Called Semantic Analysis
Goal of Semantic Analysis
• Ensure that program obe... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-sma-5502-fall-2005/41f7dfc1934579616240a609947159f0_7_semantic_check.pdf |
for v local descriptor for i
parameter descriptor for x
while (i < v.length)
v[i] = v[i]+x;
<
ldl
len
ldf
sta
ldf
ldl
+
lda
ldp
ldf
ldl
field descriptor for v local descriptor for i
parameter descriptor for x
while (i < v.length)
v[i] = v[i]+x;
while
<
ldl
len
ldf
sta
ldf
ldl
+
lda
ldp
ldf
ldl
field descriptor for ... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-sma-5502-fall-2005/41f7dfc1934579616240a609947159f0_7_semantic_check.pdf |
names
• When to check?
– when insert descriptor into local symbol table
• Parameter and field symbol tables similar
Class Descriptor
• When build class descriptor, have
– class name and name of superclass
– field symbol table
– method symbol table
• What to check?
– Superclass name corresponds to actual class
– No na... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-sma-5502-fall-2005/41f7dfc1934579616240a609947159f0_7_semantic_check.pdf |
int + double is double, float +
double is double
• Interesting oddity: C converts float procedure
arguments to doubles. Why?
Type Inference
• Infer types without explicit type declarations
• Add is very restricted case of type inference
• Big topic in recent programming language
research
– How many type declaration... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-sma-5502-fall-2005/41f7dfc1934579616240a609947159f0_7_semantic_check.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
18.969 Topics in Geometry: Mirror Symmetry
Spring 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
MIRROR SYMMETRY: LECTURE 2
DENIS AUROUX
Reference for today: M. Gross, D. Huybrechts, D. Joyce, “Calabi-Yau Mani
folds ... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/42040c1f577d7178b66e4d6e3fbd12f4_MIT18_969s09_lec02.pdf |
Ω0,1(X, E) → Ω0,2(X, E) → · · · }
∂
q(X, E) = ker∂/im∂
H
∂
∂
∂
Deforming J to a “nearby” J � gives
(4)
Ω1,0 ⊆ T ∗C = Ω1,0 ⊕ Ω0,1
J
J �
J
is a graph of a linear map (−s) : Ω1
0 (acted
,
,1 . J � is determined by Ω1
,0 → Ω0
�
J
J
J
on by i) and Ω0,1 (acted on by i�). s is a section of (Ω1,0)∗ ⊗ Ω0,1 = T1,0 ⊗ Ω... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/42040c1f577d7178b66e4d6e3fbd12f4_MIT18_969s09_lec02.pdf |
(5)
[α ⊗ v, α� ⊗ v�] = (α ∧ α�) ⊗ [v, v�]
giving it the structure of a differential graded Lie algebra.
Proposition 1. J � is integrable
∂s + 2
Proof. We want to check that the bracket of two 0, 1 tangent vectors is still 0, 1,
i.e. that
1 [s, s] = 0.
⇔
(6)
[
∂ �
+
∂zk
�
s�k
∂ �
+
∂
,
∂z� ∂zk
s�k
∂
∂z... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/42040c1f577d7178b66e4d6e3fbd12f4_MIT18_969s09_lec02.pdf |
integrable complex structures on X}/Diff(X)
(or, assuming that Aut(X, J) is discrete, we want that near J, ∃ a universal family
X → U ⊂ MCX (complex manifolds, holomorphic fibers ∼= X) s.t. any family of
integrable complex structures X � → S induces a map S → U s.t. X pulls back
to X �). We have an action of the diffe... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/42040c1f577d7178b66e4d6e3fbd12f4_MIT18_969s09_lec02.pdf |
,1(X, T X 1,0), s(0) = 00. By the above, this should satisfy
(12)
∂s(t) +
1
2
[s(t), s(t)] = 0
In particular, s1 = dt t=0 solves ∂s1 = 0. We obtain an infinitesimal action of
Diff(X): for (φt), φ0 = id , dt |t=0 = v a vector field,
dφ
|
ds
(13)
d
dt
|t=0(−(∂φt)−1 ◦ ∂φt) = −
d
dt
|t=0(∂φt) = −∂v
This implies that ... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/42040c1f577d7178b66e4d6e3fbd12f4_MIT18_969s09_lec02.pdf |
phic maps. To first order, this is given by holomorphic vector fields vij on Ui ∩ Uj
s.t. vij = −vji and vij + vjk = vik on Uijk.
Cech 1-cocycle
conditions in the sheaf of holomorphic tangent vector fields. Modding out by
holomorphic functions ψi : Ui → Ui (which act by φij �→ ψj φij ψ−1) is precisely
modding by the ˇ
... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/42040c1f577d7178b66e4d6e3fbd12f4_MIT18_969s09_lec02.pdf |
that [s1, s1] ∈ Ker (∂). Thus, the primary obstruction to deforming is the class
of [s1, s1] in H 2(X, T X 1,0). If it is zero, then there is an s2 s.t. ∂s2 + 1
2 [s1, s1] = 0,
and the next obstructure is the class of [s1, s2] ∈ H 2(X, T X 1,0). We are basically
attempting to apply by brute force the implicit functi... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/42040c1f577d7178b66e4d6e3fbd12f4_MIT18_969s09_lec02.pdf |
0) = 0. In local coordinates, we have
T ∗X 1,0 = Span{dz(t) = dzi −
t=0αt ∈ Ωp,q + Ωp+1,q−1 + Ωp−1,q+1
.
sij (t)dzj }
�
X
⇒
Jt
(17)
αt
=
i
�
I,J||I|=p,|J|=q
(
αIJ (t)dzi
1
t) ∧ · · · ∧ dzi
)
t
) ∧ · · · ∧ dz(
t
t) ∧ dz(
(
j
j
p
q
1
d
|
Taking dt t=0, the result follows from the product rule. We mostly... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/42040c1f577d7178b66e4d6e3fbd12f4_MIT18_969s09_lec02.pdf |
Massachusetts Institute of Technology
18.413: ErrorCorrecting Codes Laboratory
Professor Daniel A. Spielman
Handout 0
February 3, 2004
Signing Up
If too many people sign up for the course, I will perform a lottery among those who have signed
up, and announce the results by email on Wednesday, Feb 4th.
First Re... | https://ocw.mit.edu/courses/18-413-error-correcting-codes-laboratory-spring-2004/4215f545fb81cbc74f2849d4596c802b_out0.pdf |
System Identification
6.435
SET 9
– Asymptotic distribution of PEM
Munther A. Dahleh
Lecture 9
6.435, System Identification
1
Prof. Munther A. Dahleh
Central Limit Theorem
(Generalization)
• Basic Theorem II:
Consider
are both ARMA processes, possibly correlated, with
underlying white noise (bounded 4th moment)
Lect... | https://ocw.mit.edu/courses/6-435-system-identification-spring-2005/421a9e228140fc8e47002d404223b9bb_lec9_6_435.pdf |
are asymptotically efficient if
⇒
is normally distributed.
Estimates for accuracy
Lecture 9
6.435, System Identification
10
Prof. Munther A. Dahleh
Examples
ARX:
Lecture 9
6.435, System Identification
11
Prof. Munther A. Dahleh
MA:
Lecture 9
6.435, System Identification
12
Prof. Munther A. Dahleh
ARMA:
Lecture 9... | https://ocw.mit.edu/courses/6-435-system-identification-spring-2005/421a9e228140fc8e47002d404223b9bb_lec9_6_435.pdf |
20.110/5.60 Fall 2005
Lecture #1
page
1
Introduction to Thermodynamics
Thermodynamics:
→ Describes macroscopic properties of
equilibrium systems
→ Entirely Empirical
→ Built on 4 Laws and “simple” mathematics
0th Law ⇒ Defines Temperature (T)
1st Law ⇒ Defines Energy (U)
2nd Law ⇒ Defines Entropy (S)
3rd La... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/425a9bc953f2930faf9816e9fcfe399e_l01.pdf |
20.110/5.60 Fall 2005
Lecture #1
page
3
Two classes of Properties:
• Extensive: Depend on the size of the system
(n,m,V,…)
• Intensive: Independent of the size of the system
(T,p,
V =
V
n
,…)
The State of a System at Equilibrium:
• Defined by the collection of all macroscopic properties that
are described by ... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/425a9bc953f2930faf9816e9fcfe399e_l01.pdf |
K)
Change of State:
(Transformations)
• Notation:
3 H2 (g, 5 bar, 100 °C) = 3 H2 (g, 1 bar, 50 °C)
initial state
final state
2 H2O (ℓ, 1 bar, 50 °C) = 3 H2O (g, 1 bar, 150 °C)
initial state
final state
20.110J / 2.772J / 5.601JThermodynamics of Biomolecular SystemsInstructors: Linda G. Griffith, Kimberly Ha... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/425a9bc953f2930faf9816e9fcfe399e_l01.pdf |
the warmer to the cooler object. This continues
until they are in thermal equilibrium (the heat flow stops). At this
point, both bodies are said to have the same “temperature”.
This intuitively straightforward idea is formalized in the 0th Law of
thermodynamics and is made practical through the development of
ther... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/425a9bc953f2930faf9816e9fcfe399e_l01.pdf |
At)
Experimental result:
A = 0.0036609 = 1/273.15
-273.15 0 100 C
Note:
t
= −
273.15
°
C
(
t
= −
273.15
)
is called the absolute zero,
is special
°
C
20.110J / 2.772J / 5.601JThermodynamics of Biomolecular SystemsInstructors: Linda G. Griffith, Kimberly Hamad-Schifferli, Moungi G. Bawendi, Robert W. Field
... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/425a9bc953f2930faf9816e9fcfe399e_l01.pdf |
8.31451
J
K mol
−
(gas constant)
20.110J / 2.772J / 5.601JThermodynamics of Biomolecular SystemsInstructors: Linda G. Griffith, Kimberly Hamad-Schifferli, Moungi G. Bawendi, Robert W. Field
20.110/5.60 Fall 2005
Lecture #1
page
9
• Work:
“w”
= Fw
(cid:65)⋅
applie... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/425a9bc953f2930faf9816e9fcfe399e_l01.pdf |
Lecture 7
8.321 Quantum Theory I, Fall 2017
39
Lecture 7 (Sep. 27, 2017)
7.1 Spin Precession In a Magnetic Field
Last time, we began discussing the classic example of precession of a spin- 1 particle in a magnetic
field. The Hamiltonian of this system is
2
With B = Bzˆ, this becomes
The energy levels are
giving a level ... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/42767c0b651a33268189c7ef1f87630e_MIT8_321F17_lec7.pdf |
iωt/2c+|+(cid:105) + eiωt/2c−|−(cid:105) .
(7.7)
(7.8)
(7.9)
We now know the state of the system at all times. For example, if we initially have |ψ(cid:105) = |+(cid:105), then
which has
|ψ(t)(cid:105) = e−iωt/2|+(cid:105) ,
Prob(cid:0)Sz
(cid:126)=
(cid:1)
=2
1
(7.10)
(7.11)
for all times. This is why energy eigenstat... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/42767c0b651a33268189c7ef1f87630e_MIT8_321F17_lec7.pdf |
,
Prob(cid:0)Sx = − (cid:126)
2 at time t(cid:1) = sin2
(cid:19)
.
(cid:18) ωt
2
(7.12)
(7.13)
(7.14)
(7.15)
We can check that this is true, but we know it must be true by conserv
then have
ation of probability. We
(cid:104)Sx(cid:105) =
cos2
− sin2
(cid:20)
(cid:126)
2
(cid:19)
(cid:18) ωt
2
(cid:19)(cid:21)
(cid:18) ... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/42767c0b651a33268189c7ef1f87630e_MIT8_321F17_lec7.pdf |
(cid:126)Sy, which gives
dSz
dt
= 0 .
dSx
dt
=
iω iω
(cid:126)
e Szt/ i(cid:126)Sye−iωSzt/(cid:126) = −ωSy(t) .
(cid:126)
Similarly, using [Sz, Sy] = −i(cid:126)Sx, we find
dSy
dt
= ωSx(t) .
(7.17)
(7.18)
(7.19)
(7.20)
(7.21)
Lecture 7
8.321 Quantum Theory I, Fall 2017
41
We can write these three expression compactly i... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/42767c0b651a33268189c7ef1f87630e_MIT8_321F17_lec7.pdf |
and p commute, the corresponding operators
do not.
Suppose we have the Hamiltonian
H =
p2
2m
+ V (x) .
(7.23)
How do we proceed? In the Schr¨odinger picture, we first find the energy eigenkets |j(cid:105) and eigenvalues
Ej, which satisfy
Once we have found these eigenkets, we can expand an arbitrary state as
H|j(cid:105... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/42767c0b651a33268189c7ef1f87630e_MIT8_321F17_lec7.pdf |
17
42
From this point forward, we will drop the subscript H on the Heisenberg picture operators. Using
a commutator identity, we then have
dx
dt
=
=
=
1
i(cid:126)
1
i(cid:126)
p
m
(cid:21)
p2
2m
p
2m
x,
(cid:20)
x,
(cid:16)(cid:104)
.
(cid:105)
p + p
(cid:104)
x,
(cid:105)(cid:17)
p
2m
This is the expected result, but... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/42767c0b651a33268189c7ef1f87630e_MIT8_321F17_lec7.pdf |
.2.1 Example: Charged Particle in a Uniform Electric Field
Consider a charged particle in a uniform electric field, which has the Hamiltonian
H =
p2
m
2
− qE(t) .
x
(7.32)
In the Schr¨odinger picture, this is a messy problem to solve. In the Heisenberg picture, however,
the problem is not difficult at all. Using Eq. (7.30... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/42767c0b651a33268189c7ef1f87630e_MIT8_321F17_lec7.pdf |
dx
dt
=
p
m
,
dp
dt
=
−mω2x .
Solving these equations gives
x(t) = x(0) cos(ωt) +
p(0)
mω
sin(ωt) ,
p(t) = −mωx(0) sin(ωt) + p(0) cos(ωt) .
(7.39)
(7.40)
Finding these equations in the Schr¨odinger picture is messy, though it can be done; in the Heisenberg
picture, the result was immediate.
Keep in mind that these are ... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/42767c0b651a33268189c7ef1f87630e_MIT8_321F17_lec7.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
18.969 Topics in Geometry: Mirror Symmetry
Spring 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
MIRROR SYMMETRY: LECTURE 10
DENIS AUROUX
1. The Quintic (contd.)
Recall that we had a quintic mirror family Xˇ
ψ with L... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/42bc2e161167860681a541acc4cfc821_MIT18_969s09_lec10.pdf |
rewrite the Picard-Fuchs equation in the form
(4)
3
d4 �
ˇ
[Ω] +
dz4
k=0
ck(z)
[ ˇ
Ω] = 0
dk
dzk
1
2
DENIS AUROUX
�3
ckWk = 0. By Griffiths transversality ( dk Ω has no (0, 3)
ˇ
dzk
k=0
Then W4 +
component unless k ≥ 3), W0 = W1 = W2 = 0. Moreover,
�
d2Ωˇ
Xˇ dz2
�
�
dΩˇ
Xˇ dz
d2Ωˇ
dz2
d3Ωˇ
dz3
d... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/42bc2e161167860681a541acc4cfc821_MIT18_969s09_lec10.pdf |
2
c
1950750
c
2
1
q 2 −
2
2
c
10277490000
c
6
1
q 3 + · · ·
3
2
MIRROR SYMMETRY: LECTURE 10
3
Now we can describe the mirror symmetry: there exists a basis of H 2(X, Z) ∼= Z
(where X is the original quintic) given by the Poincar´e dual {e} of a hyperplane
s.t., writing [B + iω] = te, q = exp(2πit) = exp(2... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/42bc2e161167860681a541acc4cfc821_MIT18_969s09_lec10.pdf |
�
q
d3 n3 1 −
�
qd
n1 q 2 + 27 n3 +
8
d>0
= 5 + n1q + 8 n2 +
�
�
n1 q 3 + 64 n4 +
27
n2 +
8
�
n1 q 4 +
64
· · ·
Matching these gives
c1 = −5.
•
• n1 =
575 5
c2
· = c2
2875
: classical algebraic geometry tells us that 2875 is the
number of lines on a quintic, c2 = 1.
• n2 = 609250 (had been cal... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/42bc2e161167860681a541acc4cfc821_MIT18_969s09_lec10.pdf |
um” intersection theory involving
J-holomorphic disks. On the complex side, we look at intersections of subvarieties
and holomorphic maps/extensions of bundles/sheaves. Thus, the complex side is
governed by “classical” algebraic geometry, and all the “quantum” information
is on the symplectic side. For this, we wil... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/42bc2e161167860681a541acc4cfc821_MIT18_969s09_lec10.pdf |
λi | λi → ∞}. The Floer complex CF (L0, L1) is the free Λ-module Λ|L0∩L1|
generated by L0 ∩ L1. Our goal is to define a differential δ s.t. HF (L0, L1) =
H ∗(CF, δ) is invariant under Hamiltonian isotopies. The motivation for this was
to understand Arnold’s conjecture on Lagrangian intersections. From that point
of v... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/42bc2e161167860681a541acc4cfc821_MIT18_969s09_lec10.pdf |
Lecture 7
Burke’s Theorem and Networks of Queues
Eytan Modiano
Massachusetts Institute of Technology
Eytan Modiano
Slide 1
�Burke’s Theorem
• An interesting property of an M/M/1 queue, which greatly
simplifies combining these queues into a network, is the
surprising fact that the output of an M/M/1 queue with arriva... | https://ocw.mit.edu/courses/6-263j-data-communication-networks-fall-2002/42cb7759de031ae20aae04632773fef4_Lecture7.pdf |
, the (forward) departure process is Poisson
• By the same type of argument, the state (packets in system) left by a
(forward) departure is independent of the past departures
–
In backward process the state is independent of future arrivals
Eytan Modiano
Slide 4
NETWORKS OF QUEUES
Exponential
Exponential
Poiss... | https://ocw.mit.edu/courses/6-263j-data-communication-networks-fall-2002/42cb7759de031ae20aae04632773fef4_Lecture7.pdf |
Slow truck effect
Short packets
Long packet
queue
queue
queue
• Example of bunching from slow truck effect
long packets require long service at each node
–
– Shorter packets catch up with the long packets
• Similar to phenomenon that we experience on the roads
– Slow car is followed by many faster cars becaus... | https://ocw.mit.edu/courses/6-263j-data-communication-networks-fall-2002/42cb7759de031ae20aae04632773fef4_Lecture7.pdf |
P(n
n i
i = k
i 1
i 1
where ρi =
λi
µi
• That is, in steady state the state of node i (ni) is independent of the
states of all other nodes (at a given time)
Independent M/M/1 queues
–
– Surprising result given that arrivals to each queue are neither
Poisson nor independent
– Similar to Kleinrock’s independenc... | https://ocw.mit.edu/courses/6-263j-data-communication-networks-fall-2002/42cb7759de031ae20aae04632773fef4_Lecture7.pdf |
Tutorial #2
Verilog
Simulation
Toolflow
Tutorial Notes Courtesy of Christopher Batten
% v c s m i p s . v
s i m v
/
% .
- R P P &
% v c s
6.884 – Spring 2005
02/16/05
T02 – Verilog 1
Figure by MIT OCW.
A Hodgepodge Of Information
− CVS source management system
− Browsing a CVS repository with viewcvs
− Makef... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/42da3ce96010a59f449da886e13685c0_t02_verilog_tut.pdf |
T02 – Verilog 4
CVS Basics
Common CVS commands
– cvs checkout pname
– cvs update pname
– cvs commit [filelist] Commit your changes
– cvs add [filelist]
– cvs diff
Checkout a working copy
Update working dir vs. repos
Add new files/dirs to repos
See how working copy differs
Set the $CVSEDITOR environment vari... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/42da3ce96010a59f449da886e13685c0_t02_verilog_tut.pdf |
05
02/16/05
T02 – Verilog 7
CVS – Multiple Users
F.1
checkout
checkout
User
A
User
B
F.1
F.1
6.884 – Spring 2005
02/16/05
T02 – Verilog 8
CVS – Multiple Users
F.2
commit
User
A
User
B
F.1
F.2
6.884 – Spring 2005
02/16/05
T02 – Verilog 9
CVS – Multiple Users
F.2
commit
Conflict!
User
A
... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/42da3ce96010a59f449da886e13685c0_t02_verilog_tut.pdf |
Æ Makefiles
Why not use Makefiles to start out with?
– Dependency tracking is less necessary
– Difficult to implement some operations
Why are we changing now?
– Makefiles are more familiar to many of you
– Dependency tracking will become more useful with
the addition of test binary generation and Bluespec
compila... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/42da3ce96010a59f449da886e13685c0_t02_verilog_tut.pdf |
batten/mips2stage
% mkdir build
% cd build
% ../configure.pl ../config/mips2stage.mk
% make simv
% make self_test.bin
% make self_test.vmh
% ./simv +exe=self_test.vmh
% make run-tests
You can just use make
run-tests and the
dependency tracking will
cause the simulator and
the tests to be built
before running the... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/42da3ce96010a59f449da886e13685c0_t02_verilog_tut.pdf |
05
T02 – Verilog 19
Writing SMIPS
Assembly
You can find the
assembly format for
each instruction in the
SMIPS processor spec
next to the
instruction tables
Use self_test.S as an
example
6.884 – Spring 2005
02/16/05
T02 – Verilog 20
Writing SMIPS Assembly
Our assembler accepts three types of register
sp... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/42da3ce96010a59f449da886e13685c0_t02_verilog_tut.pdf |
1
mtc0 r2, r21
loop: beq zero, zero, loop
nop
.set reorder
TEST_CODEEND
Assembler directive which
tells the assembler not to
reorder instructions –
programmer is responsible
for filling in the delay slot
6.884 – Spring 2005
02/16/05
T02 – Verilog 23
Use smips-objdump for Disassembly
Eventually the disassem... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/42da3ce96010a59f449da886e13685c0_t02_verilog_tut.pdf |
a0000 lui $k0,0x0
101c: 275a1400 addiu $k0,$k0,5120
1020: 03400008 jr $k0
1024: 42000010 rfe
00001100 <__testexcep>:
1100: 401a6800 mfc0 $k0,$13
1104: 00000000 nop
<snip>
00001400 <__testcode>:
1400: 24010001 li $at,1
1404: 4081a800 mtc0 $at,$21
0001408 <loop>:
1408: 1000ffff b 1408 <loop>
...
141c: 3c080000 lu... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/42da3ce96010a59f449da886e13685c0_t02_verilog_tut.pdf |
408 <loop>
...
141c: 3c080000 lui $t0,0x0
1420: 8d081530 lw $t0,5424($t0)
1424: 3c01dead lui $at,0xdead
1428: 3421beef ori $at,$at,0xbeef
142c: 11010003 beq $t0,$at,143c <loop+34>
...
1438: 0000000d break
143c: 24080001 li $t0,1
1440: 4088a800 mtc0 $t0,$21
1444: 1000ffff b 1444 <loop+3c>
6.884 – Spring 2005
0... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/42da3ce96010a59f449da886e13685c0_t02_verilog_tut.pdf |
: 1 [pc=00001004] [ireg=08000500] [rd1=00000000] [rd2=00000000] [wd=00001008] tohost= 0
CYC: 2 [pc=00001400] [ireg=00000000] [rd1=00000000] [rd2=00000000] [wd=00000000] tohost= 0
CYC: 3 [pc=00001404] [ireg=24010001] [rd1=00000000] [rd2=xxxxxxxx] [wd=00000001] tohost= 0
CYC: 4 [pc=00001408] [ireg=4081a800] [rd1=xxxxx... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/42da3ce96010a59f449da886e13685c0_t02_verilog_tut.pdf |
ilog 27
Trace Output Instead of Waveforms
It is sometimes very useful to use $display calls
from the test harness to create cycle-by-cycle
trace output instead of pouring through waveforms
#include <smipstest.h>
TEST_SMIPS
TEST_CODEBEGIN
.set noat
addiu r1, zero, 1
mtc0 r1, r21
loop: beq zero, zero, loop
.set at ... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/42da3ce96010a59f449da886e13685c0_t02_verilog_tut.pdf |
[ireg=1000ffff] [rd1=00000000] [rd2=00000000] [wd=00001410] tohost= 1
CYC: 6 [pc=00001408] [ireg=00000000] [rd1=00000000] [rd2=00000000] [wd=00000000] tohost= 1
6.884 – Spring 2005
02/16/05
T02 – Verilog 28
Final Notes
Lab Assignment 1
– Don’t worry about cvs/make for now since I will be finishing
setting this u... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/42da3ce96010a59f449da886e13685c0_t02_verilog_tut.pdf |
2.997 DecisionMaking in LargeScale Systems
MIT, Spring 2004
February 17
Handout #6
Lecture Note 4
1
Averagecost Problems
In the average cost problems, we aim at finding a policy u which minimizes
Ju(x) = lim sup
1
T →∞ T
E
�
T −1
�
�
�
�
x0 = 0 .
gu(xt) �
t=0
(1)
Since the state space is finite, it c... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/42e1a355d7a74c79abb0bf9c959068e5_lec_4_v1.pdf |
., x∗, . . . . . ., x∗, . . . . . ., x∗, . . . . . . ,
� �� � � �� � � �� �
λ1
u
h(x)
λ2
u
Let Ti(x), i = 1, 2, . . . be the stages corresponding to the ith visit to state x∗, starting at state x. Let
⎡
u(x) = E ⎣
λi
t=Ti (x)
�Ti+1 (x)−1 gu(xt)
Ti+1(x) − Ti(x)
⎤
⎦
Intuitively, we must have λi
u(x), since ... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/42e1a355d7a74c79abb0bf9c959068e5_lec_4_v1.pdf |
=0
1
We can now speculate about a version of Bellman’s equation for computing λ∗ and h∗. Approximating
J ∗(x, T ) as in (3, we have
J ∗(x, T + 1) = min ga(x) +
Pa(x, y)J ∗(y, T )
�
a
λ∗(T + 1) + h∗(x) + o(T ) = min ga(x) +
Pa(x, y) [λ∗T + h∗(y) + o(T )]
�
�
�
y
�
y
�
a
�
λ∗ + h∗(x) = mina ga(x) +
�... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/42e1a355d7a74c79abb0bf9c959068e5_lec_4_v1.pdf |
T h∗ ≡ Tu∗ h∗. Then,
and
Ju∗ (x) = λ∗, ∀x,
J ∗
u∗ (x) ≤ Ju(x), ∀u.
Proof: Let u = (u1, u2, . . . ). Let N be arbitrary. Then
TuN −1 h∗ ≥ T h∗ = λ∗e + h∗
TuN −2 (h∗ + λ∗e)
TuN −2 TuN −1 h∗
≥
= TuN −2 h∗ + λ∗e
≥ T h∗ + λ∗e
= h∗ + 2λ∗e
2
Then
Thus,we have
T1T2 · · ·
TN −1h∗ ≥ N λ∗e + h∗
�
E
N −1
�
�
gu... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/42e1a355d7a74c79abb0bf9c959068e5_lec_4_v1.pdf |
exists, then the
average cost Ju∗ (x) is the same for all initial states. It is easy to come up with examples where this is not
the case. For instance, consider the case when the transition probability is an identity matrix, i.e., the state
visits itself every time, and each state incurs different transition costs g(... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/42e1a355d7a74c79abb0bf9c959068e5_lec_4_v1.pdf |
5
Properties of Linear,
Time-Invariant
Systems
In this lecture we continue the discussion of convolution and in particular ex-
plore some of its algebraic properties and their implications in terms of linear,
time-invariant (LTI) systems. The three basic properties of convolution as an
algebraic operation are that it i... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/431b597316940ea786c72a16b8cd6371_MITRES_6_007S11_lec05.pdf |
the result. The consequence of this for the intercon-
nection of LTI systems is that a parallel combination can be collapsed into a
single system whose impulse response is the sum of the two individual ones.
In looking at and understanding the algebraic properties of convolution, it is
worthwhile to recognize that conv... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/431b597316940ea786c72a16b8cd6371_MITRES_6_007S11_lec05.pdf |
arity-the fact that for a linear
system (whether or not it is time-invariant), if the input is zero for all t (or n),
then the output is zero also.
In this lecture we illustrate the properties discussed above with some
systems. The problems associated with this lecture provide the opportunity to
explore these propertie... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/431b597316940ea786c72a16b8cd6371_MITRES_6_007S11_lec05.pdf |
[k] h[n-k] =x[n] * h[n]
k=-o
Convolution Integral:
+xt
x1(t)
=f
x (r) 8 (t -T) dT
-00
+00
y(t)
=f
X(T)
h(t-T) dT = x(t)
* h(t)
-00
Commutative:
x[n] * h [n] = h[n] *x[n]
x(t) * h(t)
= h(t) *x(t)
TRANSPARENCY
5.2
The commutativity
property of
convolution.
u[n] -> au[n]
=anu[n] * u[n]
u(t) * eatu(t)
= e-atu(t) *u(t)... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/431b597316940ea786c72a16b8cd6371_MITRES_6_007S11_lec05.pdf |
parallel can
be collapsed into a
single LTI system.
Signals and Systems
MARKERBOARD
5.1
XNVEMTlI L M
p4mnpd eSPMC
I
-
)
It
0 ov ,vro o. 4 (4
->4-k
(V~ -T
~400
V,00
K0
04 oJt
1
U
'.~O
-~Or
U at
-46~
MARKERBOARD
5.2
-ov- LII
~ievs
COIIASOL I +
tV(*)O
t<0o
L\T ea'
'-t4ety
,
vx
,c \
0
0
-T
0
0
Q74' 0
+9A 6
S0
Pcc... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/431b597316940ea786c72a16b8cd6371_MITRES_6_007S11_lec05.pdf |
8tt)
-A)
kvtYO.Yrnf
_ d
LA. +) =8C4)
u. C-0= u, U
t)Ut
+
MIT OpenCourseWare
http://ocw.mit.edu
Resource: Signals and Systems
Professor Alan V. Oppenheim
The following may not correspond to a particular course on MIT OpenCourseWare, but has been
provided by the author as an individual learning resource.
For inf... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/431b597316940ea786c72a16b8cd6371_MITRES_6_007S11_lec05.pdf |
6.581J / 20.482J
Foundations of Algorithms and Computational Techniques in Systems Biology
Professor Bruce Tidor
Professor Jacob K. White
6.581 / 20.482
6.581J / 20.482J
Foundations of Algorithms and Computational Techniques in Systems Biology
Professor Bruce Tidor
Professor Jacob K. White | https://ocw.mit.edu/courses/20-482j-foundations-of-algorithms-and-computational-techniques-in-systems-biology-spring-2006/43228505195bbb23403eddae2ba590ab_l07.pdf |
MATH 18.152 COURSE NOTES - CLASS MEETING # 6
18.152 Introduction to PDEs, Fall 2011
Professor: Jared Speck
Class Meeting # 6: Laplace’s and Poisson’s Equations
We will now study the Laplace and Poisson equations on a domain (i.e. open connected subset)
Ω ⊂ Rn. Recall that
(0.0.1)
The Laplace equation is
n
def∆ = ∑
1
i=... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/4336b9d0607ac33f430908fdf4cb78ba_MIT18_152F11_lec_06.pdf |
B
1
(
)
(
J1 t, x, y, z , J2 t, x, y, z , J3 t, x, y, z
(
)
(
))
)
)
)) is
is the electric field
is the magnetic induction
the current density
Maxwell’s equations are
(1.1.1)
(1.1.2)
∂tE − ∇ ×
∇ ×
+
tB
∂
B
−
= J,
=
,
E 0
∇ ⋅ E = ρ,
=
∇ ⋅
B 0.
−
Recall that ∇× is the curl operator,
=
Let’s look for steady-state solutions... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/4336b9d0607ac33f430908fdf4cb78ba_MIT18_152F11_lec_06.pdf |
)
u z
and let f z
to e differentiable at z0 if
( ) = ( ) +
b
iv z be a complex-valued function
analysis. Let z
= +
x iy (where x, y R) b
∈
(where u, v R
a complex number,
). We recall that f is said
e
∈
exists. If the limit exists, we denote it by f
A fundamen
x0
tal result of complex analysis is the following: f is diff... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/4336b9d0607ac33f430908fdf4cb78ba_MIT18_152F11_lec_06.pdf |
iii)the solution varies continuously with the data.
Let Ω ⊂ Rn be a domain with a Lipschitz boundary, and let N denote the unit outward normal
ˆ
vector to ∂Ω. We consider the PDE
(2.0.6)
∆u(x
) = ( )
f x ,
∈
x Ω,
supplemented by some boundary conditions. The following boundary conditions are known to lead
to well-posed... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/4336b9d0607ac33f430908fdf4cb78ba_MIT18_152F11_lec_06.pdf |
Uniqueness via the Energy Method
In this section, we address the question of uniqueness for solutions to the equation (0.0.3), sup-
plemented by suitable boundary conditions. As in the case of the heat equation, we are able to
provide a simple proof based on the energy method.
Theorem 3.1. Let Ω ⊂ Rn be a smooth, bound... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/4336b9d0607ac33f430908fdf4cb78ba_MIT18_152F11_lec_06.pdf |
0 on ∂Ω, we have that w 0 in Ω, which
≡
(3.0.10)
which implies that
(3.0.11)
and we can argue as
b
∫ w∇ ˆN w dσ = −α ∫ w2 dσ
≤ 0,
∂Ω
∂
Ω
efore conclude that w 0 in Ω.
= 0,
∫ ∣∇w∣2
Ω
≡
4
MATH 18.152 COURSE NOTES - CLASS MEETING # 6
∣
∇
ˆN w ∂Ω
Now in the Neumann case, we have that
ything
w is constant in Ω. But now we ... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/4336b9d0607ac33f430908fdf4cb78ba_MIT18_152F11_lec_06.pdf |
case only; the proof is similar for other values of n. Let’s also assume
that x is the origin; as we will see, we will be able to treat the case of general x by reducing it to
on R2. For a ball of radius r, we have that
the origin. We will work with polar coordinates (r, θ)
r dθ. Note also that along ∂Br 0 , we have th... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/4336b9d0607ac33f430908fdf4cb78ba_MIT18_152F11_lec_06.pdf |
ˆ
)
=
1
2π
∫
∂B1 0
( )
∇ ( ) ( )
u
σ
σ dσ
.
ˆ
N
1
2π
∫
B1(0)
∆
u(y
) d2y.
But ∆u 0 since u is harmonic, so we have shown that
=
(4.0.17)
′
g
(r) =
0,
and we have shown (4.0.12b) for x 0.
=
T
o prove (4.0.12a), we use polar coordinate integration and (4.0.12b) (in the case x 0) to obtain
=
MATH 18.152 COURSE NOTES - CL... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/4336b9d0607ac33f430908fdf4cb78ba_MIT18_152F11_lec_06.pdf |
+ )
(
=
(4.0.12b)
2
∫
ω 2
2R BR x
)
(
x.
for general
u y d2y,
( )
(cid:3)
5. Maximum Principle
Let’s now discuss another amazing property verified by harmonic functions. The property, known
as the strong maximum principle, says that most harmonic functions achieve their maximums and
minimums only on the interior of Ω. T... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/4336b9d0607ac33f430908fdf4cb78ba_MIT18_152F11_lec_06.pdf |
)
(
u y d2
)
(
y
=
1
∣ ( )∣
B p
{∫
Br z
( )
u y d2y
( )
+ ∫
B Br z
( )
/
u y d2y
( )
}
{∣B
r
)∣
( )∣ ( ) + ∫
z
z
u
B Br z
( )
/
2
u(y) d y} ≥
1
∣ ( )∣
B p
{∣Br
(z)∣u(z) + m(∣B(p)∣ − ∣Br(z)∣)} .
Rearranging inequality (5.0.22), we conclude that
(5.0.23)
u
(z) ≤ m.
Com
bining (5.0.21) and (5.0.23), we conclude that
(5.0.... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/4336b9d0607ac33f430908fdf4cb78ba_MIT18_152F11_lec_06.pdf |
to the data f, g C Ω , then
inciple
∈
(1) (Comparison Pr
2
) If f
≥ g on
∂Ω and f ≠ g, then
uf
>
ug in Ω.
(2) (Stability Estimate) For any x
∈
Ω, we have that
∣
uf x
( ) −
u
( )∣ ≤
g x
( )
∣
max f y
y ∂Ω
∈
−
)∣
(
g y .
Proof. We first prove the Comparison
we see that w solves
Principle.
Let
w
=
u
f
−
ug
. Then by subtra... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/4336b9d0607ac33f430908fdf4cb78ba_MIT18_152F11_lec_06.pdf |
the case f
=
t of the corollary now follows directly from applying the Stability
(cid:3)
MIT OpenCourseWare
http://ocw.mit.edu
18.152 Introduction to Partial Differential Equations.
Fall 2011
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/4336b9d0607ac33f430908fdf4cb78ba_MIT18_152F11_lec_06.pdf |
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