text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
y satisfy the
s 2 = 1, sx = −xs, sˆ
y = −ˆ yx − xˆ
ys, ˆ
y = 1 − 2cs.
The algebra generated by s, x, yˆ with these relations is the rational Cherednik algebra
H1,c(Z2, C) with the action of Z2 on C is given by z → −z.
7.12. Affine and extended affine Weyl groups. Let R =
be a root system
with respect to a nondegenerate... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
, j] ∈ Rn × R| where α ∈ R, j ∈ Z}.
The set of positive affine roots is Ra = {[α, j]
[−θ, 1]. We will identify α ∈ R with ˜α = [α, 0] ∈ Ra .
+
| j ∈ Z>0} ∪ {[α, 0] | α ∈ R+}.
Define α0 =
For an arbitrary affine root ˜α = [α, j] and a vector ˜z = [z, ζ] ∈ Rn × R, the corresponding
affine reflection is defined as follows: ... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
W
a
a permuting simple affine roots under their action in Rn+1 . It is a normal
the elements of W
commutative subgroup of Aut = Aut(Dyna) (Dyna denotes the affine Dynkin diagram).
The quotient Aut/Π is isomorphic to the group of the automorphisms preserving α0, i.e. the
group AutDyn of automorphisms of the finite Dynki... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
can define an action on these X[b,j] by ˆwX[b,j] = Xwˆ[b,j].
Definition 7.24 (Cherednik). The double affine Hecke algebra (DAHA) of the root system
R, denoted by HH, is an algebra defined over the field Cq,t = C(q1/m, t1/2), generated by
Ti, i = 0, . . . , n, Π, Xb, b ∈ P , subject to the following relations:
i
(1) TiTj ... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
�)yi + u
ext
generated by the group algebra of W a , Π and pairwise commutative y˜b =
for ˜b = [b, u] ∈ P × Z, with the following relations:
� n
siyb − ysi(b)si = −ki(b, αi
∨), for i = 1, . . . , n,
s0yb − ys0(b)s0 = k0(b, θ),
πrybπr
−1 = yπr (b) for πr ∈ Π.
Remark 7.25. This degeneration can be obtained from the... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
i, j) such that mij < ∞ and defining
relations
mij
aij = 1.
aij
tij,k = t−1
ji,−k,
2 = 1,
si
mij
�
[tij,k, ti�j�,k� ] = 0,
sptij,k = tji,ksp,
(sisj − tij,k) = 0 if mij < ∞.
k=1
Notice that if we set tij,k = exp(2πki/mij ), we get C[W ].
59
Define also the algebra A+(W ) over R := C[tij,k] (tij,k = t−1 ) by g... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
Proof of (iii), which is
quite nontrivial, can be found in [ER] (it uses the geometry of constructible sheaves on the
Coxeter complex of W ).
Let us write the relation
as a deformed braid relation:
mij
�
(sisj − tij,k) = 0
k=1
sj sisj . . . + S.L.T. = tij sisj si . . . + S.L.T.,
where tij = (−1)mij +1tij,1 · ·... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
is a “deformation” of C[W+] over R, and similarly A(W ) is a “twisted
deformation” of C[W ].
Now let Γ = Γ(m1, . . . , mr), r ≥ 3, be the Fuchsian group defined by generators cj ,
j = 1, . . . , r, with defining relations
c m
j
j = 1,
r
�
cj = 1.
Here 2 ≤ mj < ∞.
j=1
60
�
Suppose Γ acts on H where H is a simpl... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
mi,i+1 := mi
(i ∈ Z/rZ), and mij = ∞ otherwise. Suppose the corresponding Coxeter group is W . Then
we can see that Γ = W+. Notice that the algebra Hτ (Γ, H) for genus 0 orbifolds is the
algebra A+(W ), i.e., we have Hτ (Γ, H) = A+(W ).
The condition
(1 − 1/mj ) ≥ 2 is equivalent to the condition that W has no fini... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
3.044 MATERIALS PROCESSING
LECTURE 4
General Heat Conduction Solutions:
∂T = ∇ · k∇T, T (¯x, t)
∂t
Trick one: steady state ∇2T = 0, T (x)
Trick two: low Biot number ∂T = α h(Ts − Tf ), T (t)
∂t
Low Biot Number Solutions: Newtonian Heating / Cooling
Global Heat Balance:
qconv = qlost
A h(T − Tf ) = −ρ cp
∂T
... | https://ocw.mit.edu/courses/3-044-materials-processing-spring-2013/02f79fb702ece65067f5827044029807_MIT3_044S13_Lec04.pdf |
1 − T0
T − T0
T1 − T0
(−1)
= erfc
(cid:18)
= erfc
(cid:18)
(cid:19)
x
√
α t
x
(cid:19)
√
α t
2
2
(−1)
4
LECTURE 4
− (T − T0)
T0 − T
T1 − T0
T − T1
T1 − T0
T − T1
T0 − T1
(cid:19)
(cid:18)
x
= erf √
2 α t
x
(cid:18)
= −erf √
2 α t
x
(cid:19)
(cid:18)
= erf √
2 α t
(cid:19)
(cid... | https://ocw.mit.edu/courses/3-044-materials-processing-spring-2013/02f79fb702ece65067f5827044029807_MIT3_044S13_Lec04.pdf |
http://ocw.mit.edu
3.044 Materials Processing
Spring 2013
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/3-044-materials-processing-spring-2013/02f79fb702ece65067f5827044029807_MIT3_044S13_Lec04.pdf |
Finite Element Modeling of the
Detachment of Soft Adhesives
Stick-slip phenomena and Schallamach waves
captured using reversible cohesive elements
1 Evelyne Ringoot
•
•
•
•
BSc in Engineering Sciences at VUB Brussels 2018
Msc in Civil Engineering at VUB Brussels 2020
o
Specialization in geomechanics and numerical met... | https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-summer-2020/03385313c1d67aae2fd49bdd6d040b32_MIT18_085Summer20_lec_ER.pdf |
plab (2014), Worm-Inspired Glue Mends Broken Hearts, consulted on Sept 2020 on https://www.karplab.net/portfolio-item/worm-inspired-glue-mends-broken-hearts
4 Experimental
observations
Research questions in
mechanics of solids: how to
explain, predict and influence
physical realities?
Analytical theory
Cohesive ele... | https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-summer-2020/03385313c1d67aae2fd49bdd6d040b32_MIT18_085Summer20_lec_ER.pdf |
MORE MATLAB INSTRUCTIONS
1. Plotting functions
In this section, the basics of plotting functions in MATLAB are described. Throughout we work
with the example of two functions, f (t) = (2t + 1)e−t sin(t), and g(t) = (t − 1)e−t cos(t).
Step 1, Specify the domain: Functions are defined on an interval called the domain.... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/034106b06ab0a11998263a17e7eeb5d7_matlab2.pdf |
define functions, and
the corresponding syntax in MATLAB. In the list, y(t) and z(t) are names for functions or pieces
of functions that are already specified.
Operation
y(t) + z(t)
y(t)z(t)
y(t)n
y(t)/z(t)
sin(y(t))
cos(y(t))
ey(t)
ln(y(t))
log10(y(t))
MATLAB Syntax
y + z
y.* z
y.^n
y./z
sin(y)
cos(y)... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/034106b06ab0a11998263a17e7eeb5d7_matlab2.pdf |
plot a
parametrized figure. For instance, for the parametrized curve (y, z) where y(t) = (2t + 1)e−t sin(t),
z(t) = (t − 1)e−t cos(t), the syntax is,
1
>> i = plot(y,z)
where y and z are specified as above. Note that when plotting parametrized curves, it is still
necessary to specify the tdomain. But t doesn’t ex... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/034106b06ab0a11998263a17e7eeb5d7_matlab2.pdf |
a JPEG file, click
on the “File” button of the new window and then click on “Print” or “Export” in the popup menu.
There are other extras that you can find out by experimenting (such as adding labels to your axes).
2 | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/034106b06ab0a11998263a17e7eeb5d7_matlab2.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
18.306 Advanced Partial Differential Equations with Applications
Fall 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Discrete to Continuum Modeling.
Rodolfo R. Rosales
.
(cid:3)
MIT, March, 2001.
Abstract
These notes giv... | https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/0356741cb05f4800faa412e2d9bc4b84_MIT18_306f09_lec25_Discrete_to_Contin.pdf |
. . . . . .
6
| Long Wave Limit
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Transversal Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Stability of the Equilibrium Solutions . . . . . . . . . . . . . . . . . . . . . . . . . .
7
Nonlinear Elastic Wave Equat... | https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/0356741cb05f4800faa412e2d9bc4b84_MIT18_306f09_lec25_Discrete_to_Contin.pdf |
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
| Sine-Gordon Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
| Boundary Conditions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
Kinks and Breathers for the Sine Gordon Equation . . . . . . . . . . . . . .... | https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/0356741cb05f4800faa412e2d9bc4b84_MIT18_306f09_lec25_Discrete_to_Contin.pdf |
, near every point in space and at every
time, by this con(cid:12)guration | for some value of the parameters. The parameters are then assumed
to vary in space and time, but on scales (macro-scales) that are much larger than the ones associated
with the basic con(cid:12)guration (micro-scales). Then one attempts to der... | https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/0356741cb05f4800faa412e2d9bc4b84_MIT18_306f09_lec25_Discrete_to_Contin.pdf |
)
f
(x
x
) ;
(2.1)
+1
1
n
n
+
n
n
n
n(cid:0)
2
n
n(cid:0)
2
2
1
1
dt
(cid:0)
(cid:0)
(cid:0)
for n = 0;
1;
2; : : : The simplest solution for this system of equations is equilibrium.
In this
case all the accelerations vanish, so that the particle positions are given by the series of algebraic
(cid:6)
(cid:6)
By some de... | https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/0356741cb05f4800faa412e2d9bc4b84_MIT18_306f09_lec25_Discrete_to_Contin.pdf |
;
(2.3)
+
+
n
n
2
2
L
(cid:18)
(cid:19)
where F
is a non-dimensional mathematical function, of O(1) size, and with O(1) deriva-
+
n
2
1
tives. A further assumption is that F
changes slowly with n, so that two nearby springs
+
n
2
1
are nearly equal. Mathematically, this is speci(cid:12)ed by stating that:
1
F
((cid:17)... | https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/0356741cb05f4800faa412e2d9bc4b84_MIT18_306f09_lec25_Discrete_to_Contin.pdf |
12)ed time
n
n
scale (cid:28) to non-dimensionalize time, namely:
t = (cid:28) T ; the equations become:
2
2
d
(cid:15) f (cid:28)
X
X
X
X
+1
1
n
n
n
n(cid:0)
1
1
M ((cid:15)n)
X
=
F
F
:
(2.7)
n
+
(cid:0)
(cid:0)
2
2
2
n
n(cid:0)
dT
m L
(cid:15)
(cid:15)
(cid:0)
(cid:18)
(cid:18)
(cid:19)
(cid:18)
(cid:19)(cid:19)
A an... | https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/0356741cb05f4800faa412e2d9bc4b84_MIT18_306f09_lec25_Discrete_to_Contin.pdf |
); t) + O((cid:15)
) ;
(cid:15)
@ s
2
(cid:15)
@ s
2
(cid:0)
with a similar formula applying to the di(cid:11)erence F
F
.
1
1
+
n
n(cid:0)
2
2
(cid:0)
Equation (2.9) suggests that we should take
m L
(cid:28) =
;
(2.10)
2
s
(cid:15)
f
for the un-speci(cid:12)ed time scale in (2.7). Then equation (2.9) leads to the cont... | https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/0356741cb05f4800faa412e2d9bc4b84_MIT18_306f09_lec25_Discrete_to_Contin.pdf |
equal, then we can take M
1 and F is independent of
s. Furthermore, if we take L to be the (common) equilibrium length of the springs, we then have
(cid:17)
2
2
@
@
@
@
@
2
X =
F
X
= c
X
X ;
(2.12)
2
2
@T
@ s
@ s
@ s
@ s
!
!
2
2
where c
= c
((cid:17) ) = dF =d(cid:17) ((cid:17) ) > 0, and F (1) = 0 (equilibrium length)... | https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/0356741cb05f4800faa412e2d9bc4b84_MIT18_306f09_lec25_Discrete_to_Contin.pdf |
m L
(cid:15)(cid:28)
This corresponds to a time scale much shorter than the one involved in the solution in (2.8{2.11).
What role do the motions in these scales play in the behavior of the solutions of (2.1), under the
assumptions made earlier in A and B?
For real crystal lattices, which are de(cid:12)nitely not one di... | https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/0356741cb05f4800faa412e2d9bc4b84_MIT18_306f09_lec25_Discrete_to_Contin.pdf |
sin
and
< (cid:20) <
is a constant.
(2.18)
n
(cid:20)
(cid:0)
(cid:6)
(cid:0) 1
1
2
(cid:18)
(cid:19)
These must be added to an equilibrium solution x
= (cid:11) L n = s
, where (cid:11) > 0 is a constant.
n
n
4
Check that these are solutions.
Discrete to Continuum Modeling.
6
MIT, March, 2001 | Rosales.
Relative to t... | https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/0356741cb05f4800faa412e2d9bc4b84_MIT18_306f09_lec25_Discrete_to_Contin.pdf |
this long wave limit we see that (2.19) implies that the solutions
(cid:29)
have the same wave speed c
=
c. This corresponds to the situation in (2.13 { 2.14).
w
(cid:6)
It is clear that, in the linear lattice situation described above, we cannot dismiss the fast vibration
excitations (with frequencies of the order of ... | https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/0356741cb05f4800faa412e2d9bc4b84_MIT18_306f09_lec25_Discrete_to_Contin.pdf |
;
(2.20)
n
n
+
+
(cid:0)
(cid:0)
2
n
n
n(cid:0)
n(cid:0)
2
2
2
2
1
1
dt
(cid:1)r
(cid:0)
(cid:1)r
+
n
2
2
n(cid:0)
for n = 0;
1;
2; : : : (you should convince yourself that this is the case).
(cid:6)
(cid:6)
The simplest solution for this system of equations is equilibrium, with all the masses lined up
horizontally y
=... | https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/0356741cb05f4800faa412e2d9bc4b84_MIT18_306f09_lec25_Discrete_to_Contin.pdf |
x
= nL, in
n
fact X plays here the same role that s played in the prior derivation
), and T = t=(cid:28) , where (cid:28) is as
7
in (2.10). Then the continuum limit for the equations in (2.20) is given by
2
@
Y
@
F (X;
)
@Y
M (X )
=
(2.21)
S
2
@T
@X
@X
!
where Y = Y (X; T ) and
S
2
@Y
=
1 +
:
v
S
u
@X
!
u
t
The deriva... | https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/0356741cb05f4800faa412e2d9bc4b84_MIT18_306f09_lec25_Discrete_to_Contin.pdf |
:
(2.23)
S
2
@T
@X
@X
!
S
Next, for smal l disturbances we have
1, and (2.23) can be approximated by the linear wave
equation
2
S (cid:25)
2
Y
= c
Y
;
(2.24)
T T
XX
where c
= F (1) is a constant (see equations (2.13 { 2.14).
7
The coordinate
is simply a label for the masses. Since in this case the masses do not move ho... | https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/0356741cb05f4800faa412e2d9bc4b84_MIT18_306f09_lec25_Discrete_to_Contin.pdf |
vibration for a string are obtained. Since strings have no bending strength,
these equations will be well behaved only as long as the string is under tension everywhere.
Bending strength is easily incorporated into the mass-spring chain model. Basically, what we need
to do is to incorporate, at the location of each mas... | https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/0356741cb05f4800faa412e2d9bc4b84_MIT18_306f09_lec25_Discrete_to_Contin.pdf |
that its suspending
n
n
rod makes with the vertical direction. Each pendulum is then sub ject to three forces:
(a) Gravity, for which only the component perpendicular to the pendulum rod is considered.
9
(b) Axle torsional force due to the twist (cid:18)
(cid:18)
. This couples each pendulum to the next one.
+1
n
n
(c)... | https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/0356741cb05f4800faa412e2d9bc4b84_MIT18_306f09_lec25_Discrete_to_Contin.pdf |
1
‘
The (cid:12)rst and last pendulum are at a distance
from the respective ends of the axle.
(cid:15)
2(N + 1)
The tangential force (perpendicular to the pendulum rod) due to gravity on each of the masses is
1
F
=
M g sin (cid:18)
; where n = 1; : : : ; N :
(3.3)
g
n
(cid:0)
N
For any two successive masses, there is a... | https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/0356741cb05f4800faa412e2d9bc4b84_MIT18_306f09_lec25_Discrete_to_Contin.pdf |
)
+
((cid:18)
(cid:18)
)
((cid:18)
(cid:18)
) ;
(3.6)
+1
1
n
n
n
n(cid:0)
‘L
‘L
(cid:0)
(cid:0)
(cid:0)
for n = 2; : : : ; N
1; and
(cid:0)
2
1
d
(cid:18)
1
(N + 1) (cid:20)
N
M L
=
M g sin (cid:18)
((cid:18)
(cid:18)
) :
(3.7)
1
N
N
N (cid:0)
2
N
dt
N
‘L
(cid:0)
(cid:0)
(cid:0)
These are the equations for N torsion co... | https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/0356741cb05f4800faa412e2d9bc4b84_MIT18_306f09_lec25_Discrete_to_Contin.pdf |
n +
1
2
(cid:1)x = x
x
=
:
(3.9)
+1
n
n
(cid:0)
N + 1
‘
Take equation (3.6), and multiply it by N=‘. Then we obtain
2
d
(cid:18)
N (N + 1)(cid:20)
n
(cid:26) L
=
(cid:26) g sin (cid:18)
+
((cid:18)
2(cid:18)
+ (cid:18)
) ;
+1
1
n
n
n
n(cid:0)
2
2
dt
‘
L
(cid:0)
(cid:0)
where (cid:26) = M =‘
is the mass density per unit... | https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/0356741cb05f4800faa412e2d9bc4b84_MIT18_306f09_lec25_Discrete_to_Contin.pdf |
(cid:0)
g
(cid:20)
where ! =
is the pendulum angular frequency, and c =
is a wave propagation speed
2
L
(cid:26)L
s
r
(check that the dimensions are correct).
Remark 3.4 Boundary Conditions.
What happens with the (cid:12)rst (3.5) and last (3.7) equations in the limit N
?
! 1
As above, multiply (3.5) by 1=‘. Then the e... | https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/0356741cb05f4800faa412e2d9bc4b84_MIT18_306f09_lec25_Discrete_to_Contin.pdf |
ferromagnetic materials of waves carrying rotations in the magnetization
direction, etc.
Mathematically, it is a very interesting because it is one of the few physically
10
important nonlinear partial di(cid:11)erential equations that can be solved explicitly (by a
technique known as Inverse Scattering, which we will n... | https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/0356741cb05f4800faa412e2d9bc4b84_MIT18_306f09_lec25_Discrete_to_Contin.pdf |
as kinks or anti-kinks (depending
on the sign of the rotation), and can be written explicitly. In fact, they are steady wave solutions,
11
for which the equation reduces to an O.D.E., which can be explicitly solved.
Let
1 < c < 1 be a constant (kink, or anti-kink speed), and let z = (x
c t
x
) be a moving
0
(cid:0)
(ci... | https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/0356741cb05f4800faa412e2d9bc4b84_MIT18_306f09_lec25_Discrete_to_Contin.pdf |
for nuclear inter-
(cid:15)
actions. In this interpretation, the kinks are nuclear particles. Since (in the non-dimensional
version (3.12)) the speed of light is 1, the restriction
1 < c < 1 is the relativistic restriction,
and the factor (cid:12) incorporates the usual relativistic contraction.
(cid:0)
The anti-kink s... | https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/0356741cb05f4800faa412e2d9bc4b84_MIT18_306f09_lec25_Discrete_to_Contin.pdf |
space. These solutions vanish (exponential ly) as x
.
This last property al lows for easy numerical simulations of interactions of breathers (and kinks).
! (cid:6)1
One can setup initial conditions corresponding to the interaction of as many kinks and/or breathers
as one may wish (limited only be the numerical resoluti... | https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/0356741cb05f4800faa412e2d9bc4b84_MIT18_306f09_lec25_Discrete_to_Contin.pdf |
period . . . . . . . . . . . . . . T
= 2 (cid:25)=(B C ) ;
(3.18)
p
Phase.
wave-length . . . . . . . . . (cid:21)
= 2 (cid:25)=(B C V ) ;
p
9
>
=
>
;
speed . . . . . . . . . . . . . . c
=
V ,
e
Envelope.
(3.19)
width . . . . . . . . . . . . . .
e
)
(cid:21)
= 2 (cid:25)=(d B ) ;
Notice that, while the phase moves faste... | https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/0356741cb05f4800faa412e2d9bc4b84_MIT18_306f09_lec25_Discrete_to_Contin.pdf |
ly (and accurately) on the "Fourier Side" | since there it
amounts to term by term multiplication of the n
Fourier coeÆcient by in. On the other hand,
th
non-linear operations (such as calculating the square, point by point, of the solution) can be done
eÆciently on the "Physical Side".
Thus, in a numerical computation... | https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/0356741cb05f4800faa412e2d9bc4b84_MIT18_306f09_lec25_Discrete_to_Contin.pdf |
U = e
: Then
iu
gives a formula for u
that ignores 2(cid:25) jumps in u. Warning: In the actual implementation one
xx
2
(U
)
U U
x
xx
u
= i
(3.21)
xx
(cid:0)
2
U
must use
2
(U
)
U U
x
xx
u
=
imag
xx
(cid:0)
2
(cid:0)
U
!
to avoid smal l imaginary parts in the answer (caused by numerical errors).
4 Suggested problems.
A... | https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/0356741cb05f4800faa412e2d9bc4b84_MIT18_306f09_lec25_Discrete_to_Contin.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
8.512 Theory of Solids II
Spring 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Lecture 6: Scaling Theory of Localization
6.1 Notion of dimensionless conductance.
g =
G
e2/π�
G = σLd−2
where G is (resistance)−1
L =... | https://ocw.mit.edu/courses/8-512-theory-of-solids-ii-spring-2009/039ae328a78eb6921c16239e587e69d0_MIT8_512s09_lec06.pdf |
d−2 =
2e2 �D
·
�
L2
· N (0)
1
(1)
(2)
2
(3)
6.2 Thouless Energy
The last equation suggests to define a quantity called Thouless Energy, ET
ET =
�D
L2
=
�
τT
N (0) ∼
1
Δ
· · ·
Δ is level spacing
⇒
G ∼
2e2 ET
� Δ
⇒
g ∼
ET
Δ
Now,
Physical interpretation
Assume the one box problem is solved and... | https://ocw.mit.edu/courses/8-512-theory-of-solids-ii-spring-2009/039ae328a78eb6921c16239e587e69d0_MIT8_512s09_lec06.pdf |
properties.
3
6.3 Sensitivity of energy eigenvalues to boundary conditions and its relation to ET .
Here we derive another form of ET . Usually one uses periodic boundary conditions
ψα(x + L) = ψα(x)
· · ·
ψα is an eigenstate of the Hamiltonian.
One can as well use the twisted boundary condition
ψα(x + L) = e ... | https://ocw.mit.edu/courses/8-512-theory-of-solids-ii-spring-2009/039ae328a78eb6921c16239e587e69d0_MIT8_512s09_lec06.pdf |
o
� �
= ψ�(x) exp [iφB]
�
where φB is magnetic flux. We will compute ΔEα using perturbation theory.
φ
L
To first order in H �, ΔEα = 0 as < H � >∼< Vx >= 0.
H = VxA = Vx
e
c
�
For perturbation theory second order in A, we have two terms: the diamagnetic term
which gives a constant shift to all energy levels
e2 ... | https://ocw.mit.edu/courses/8-512-theory-of-solids-ii-spring-2009/039ae328a78eb6921c16239e587e69d0_MIT8_512s09_lec06.pdf |
on Eq.(4). On general grounds, there is equal probability
for any level to go up or down and so average variation in Eα’s should be zero. By using
Thomas-Reich-Kunz F -sum rule, one can indeed show that
∂2Eα = 0
2∂φ
α
�
5
Now, | < β|Vx|α > |2 ∼ V 2 . Fluctuation of such terms is dominated by the term with
ma... | https://ocw.mit.edu/courses/8-512-theory-of-solids-ii-spring-2009/039ae328a78eb6921c16239e587e69d0_MIT8_512s09_lec06.pdf |
states and hence small conductance. Thus we have established the
relevance of Thouless energy for conductance.
6.4 Application to 1 − Δ System
6
One expects Ohm’s low behavior for small length.
g(L) ∼
1
L
as L increases, g(L) decreases
as
and so the states are localized.
g(L) < 1 ⇒
ET
Δ
< 1
⇒
... | https://ocw.mit.edu/courses/8-512-theory-of-solids-ii-spring-2009/039ae328a78eb6921c16239e587e69d0_MIT8_512s09_lec06.pdf |
low temperature as Lφ increases with
decrease in temperature.
Thus one expects the following behavior of conductivity of a one-dimensional system.
For Lφ > ξ, we can think of conductivity in terms of hopping of electrons over a distance
scale ξ in time τφ. This gives rise to a diffusion constant of ξ2/τφ and resisti... | https://ocw.mit.edu/courses/8-512-theory-of-solids-ii-spring-2009/039ae328a78eb6921c16239e587e69d0_MIT8_512s09_lec06.pdf |
Lecture 3
Properties of MLE: consistency,
asymptotic normality.
Fisher information.
In this section we will try to understand why MLEs are ’good’.
Let us recall two facts from probability that we be used often throughout this course.
•
Law of Large Numbers (LLN):
If the distribution of the i.i.d. sample X1, . .... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/03b407da8a94b3fe22d987453807ca46_lecture3.pdf |
words, the random variable ≥n(X¯n −
from normal distribution when n gets large.
EX1) will behave like a random variable
Exercise. Illustrate CLT by generating 100 Bernoulli random varibles B(p) (or one
EX1). Repeat this many times
Binomial r.v. B(100, p)) and then computing ≥n(X¯n −
and use ’dfittool’ to see that ... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/03b407da8a94b3fe22d987453807ca46_lecture3.pdf |
−0.6
−0.8
−1
�(ϕ)
PSfrag replacements
0
0.5
1
1.5
2
ϕˆ
2.5
3
3.5
4
ϕ
Figure 3.1: Maximum Likelihood Estimator (MLE)
Suppose that the data X1, . . . , Xn is generated from a distribution with unknown pa
rameter ϕ0 and ϕˆ is a MLE. Why ϕˆ converges to the unknown parameter ϕ0? This is not
immediately ob... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/03b407da8a94b3fe22d987453807ca46_lecture3.pdf |
need the
following
Lemma. We have that for any ϕ,
Moreover, the inequality is strict, L(ϕ) < L(ϕ0), unless
L(ϕ)
≡
L(ϕ0).
P�0 (f (X ϕ) = f (X ϕ0)) = 1.
|
|
which means that P� = P�0 .
Proof. Let us consider the difference
L(ϕ)
−
L(ϕ0) = E�0 (log f (X
log f (X
ϕ)
|
−
ϕ0)) = E�0 log
|
Since log t
t
−
≡
1,... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/03b407da8a94b3fe22d987453807ca46_lecture3.pdf |
consistent, i.e. ϕˆ
ϕ0 as n
.
� →
�
The statement of this Theorem is not very precise but but rather than proving a rigorous
mathematical statement our goal here is to illustrate the main idea. Mathematically inclined
students are welcome to come up with some precise statement.
Ln(ϕ)
L(ϕ)
PSfrag replacements
ˆϕ... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/03b407da8a94b3fe22d987453807ca46_lecture3.pdf |
to
ϕ.
|
|
|
|
|
Definition. (Fisher information.) Fisher information of a random variable X with
distribution P�0 from the family
I(ϕ0) = E�0 (l�(X
P� : ϕ
{
∞
ϕ0))2
|
is defined by
�
}
�
E�0 �ϕ
�
�
log f (X
19
2
.
ϕ)
|
�=�0
�
�
�
�
Remark. Let us give a very informal interpretation of Fisher informatio... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/03b407da8a94b3fe22d987453807ca46_lecture3.pdf |
��cult to distinguish, so our estimation will be worse. We will see
precisely this behavior in Theorem below.
Next lemma gives another often convenient way to compute Fisher information.
Lemma. We have,
E�0 l��(X
ϕ0)
|
�
�2
E�0 �ϕ2 log f (X
ϕ0) =
|
I(ϕ0).
−
Proof. First of all, we have
ϕ) = (log f (X
l�(X
|
... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/03b407da8a94b3fe22d987453807ca46_lecture3.pdf |
(x
f
)
|
�
E�0 (l�(X
0
−
=
=
�
�
(log f (x ϕ0))��f (x ϕ0)dx
|
|
2
f (x
ϕ0)dx
|
�
�
ϕ0))2 = 0
|
I(ϕ0 =
I(ϕ0).
−
−
20
We are now ready to prove the main result of this section.
Theorem. (Asymptotic normality of MLE.) We have,
≥n(ϕˆ
ϕ0)
−
�
N
0,
�
1
I(ϕ0)
�
.
As we can see, the asymptot... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/03b407da8a94b3fe22d987453807ca46_lecture3.pdf |
and ≥n(ϕˆ
ϕ0) =
−
≥nL
(ϕ0)
�
n
.
ˆ
Ln�� ϕ1)
(
−
(3.0.1)
Since by Lemma in the previous section we know that ϕ0 is the maximizer of L(ϕ), we have
L�(ϕ0) = E�0 l�(X ϕ0) = 0.
(3.0.2)
|
Therefore, the numerator in (3.0.1)
≥nLn� (ϕ0) = ≥n
�
= ≥n
1
n
1
n
n
i=1
�
n
l�(Xi|
ϕ0)
−
l�(Xi|
ϕ0)
−
�
E�0 l�(X1|
�
... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/03b407da8a94b3fe22d987453807ca46_lecture3.pdf |
3.0.3) we get
≥nL�n(ϕ0)
−
L��n(ϕˆ1) �
d N
0,
�
Var�0 (l�(X1|
(I(ϕ0))2
ϕ0))
.
�
Finally, the variance,
Var�0 (l�(X1|
ϕ0)) = E�0 (l�(X
ϕ0))2
|
(E�0 l�(x
ϕ0))2 = I(ϕ0)
|
−
−
0
where in the last equality we used the definition of Fisher information and (3.0.2).
Let us compute Fisher information for some particul... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/03b407da8a94b3fe22d987453807ca46_lecture3.pdf |
) has p.d.f.
≥n(ˆp
−
p0)
�
N(0, p0(1
−
p0))
f (x �) =
|
�
�e−
0,
�x , x
0
∀
x < 0
and, therefore,
−
This does not depend on X and we get
log f (x
�) = log �
|
�x
≤
�2
��2 log f (x
|
�) =
1
− �2 .
1
.
�2
Therefore, the MLE ˆ� = 1/X¯ is asymptotically normal and
�) =
|
log f (X
I(�) =
�2
��2... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/03b407da8a94b3fe22d987453807ca46_lecture3.pdf |
8.701
0. Introduction
0.6 Particles
Introduction to Nuclear
and Particle Physics
Markus Klute - MIT
1
Introduction
10-10m
a few 10-15m
10-15m
1fm
2
Force Particles
© Mattson Rosenbaum (on PBworks). All rights
reserved. This content is excluded from our
Creative Commons license. For more
information, see https://oc... | https://ocw.mit.edu/courses/8-701-introduction-to-nuclear-and-particle-physics-fall-2020/03b975f5fefb9265404c165115fa4206_MIT8_701f20_lec0.6.pdf |
from our Creative Commons
license. For more information, see https://ocw.mit.edu/fairuse.
Nuclei
Bound state of protons and neutrons
through the strong force.
Can be described by number of protons,
Z, (atomic number) and number of
neutrons, N. The sum Z+N is denoted
atomic mass A
Courtesy of Sjlegg on Wikimedia.... | https://ocw.mit.edu/courses/8-701-introduction-to-nuclear-and-particle-physics-fall-2020/03b975f5fefb9265404c165115fa4206_MIT8_701f20_lec0.6.pdf |
Turbulent Flow and Transport
1
Review of Fundamental Laws and Constitutive
Equations
1.1
Fundamental laws governing continuum flow, expressed in terms of (i) material
volumes (closed systems) and (ii) control volumes.
1.2 Mass conservation equation; integral and differential forms.
The equation of motion in terms of... | https://ocw.mit.edu/courses/2-27-turbulent-flow-and-transport-spring-2002/03ca7e90be656ad8f628a6400036aa62_Fundamentals.pdf |
nd ed. (1991): 59−89, 96−100; Pope. Ch. 2; or other books.
Secs 1.4 − 1.6: Class notes plus summaries handed out by Sonin.
Sec. 1.10:
Sec. 1.11:
For Sec. 1.7−1.8: Handout: Sonin."The Thermodynamic Constitutive
Equations and the Equation for Temperature."
Class notes.
See for example Bird, Curtis, and Hirschfelder. Mol... | https://ocw.mit.edu/courses/2-27-turbulent-flow-and-transport-spring-2002/03ca7e90be656ad8f628a6400036aa62_Fundamentals.pdf |
6.001 Structure and Interpretation of Computer Programs. Copyright © 2004 by Massachusetts Institute of Technology.
6.001 Notes: Section 7.1
Slide 7.1.1
In the past few lectures, we have seen a series of tools for
helping us create procedures to compute a variety of
computational processes. Before we move on to mo... | https://ocw.mit.edu/courses/6-001-structure-and-interpretation-of-computer-programs-spring-2005/03e0268a995c2575b1039dd867c8068b_lecture7webhand.pdf |
thinks naturally of a vector as a pairing of an x and y coordinate. One wants to
6.001 Structure and Interpretation of Computer Programs. Copyright © 2004 by Massachusetts Institute of Technology.
be able to get out the coordinates when needed, but in many cases, one thinks naturally of manipulating a vector as
a u... | https://ocw.mit.edu/courses/6-001-structure-and-interpretation-of-computer-programs-spring-2005/03e0268a995c2575b1039dd867c8068b_lecture7webhand.pdf |
– and in fact you have already seen one such example, our
implementation ofsqrt. When we implemented our
method for square roots, we actually engaged in many of these
stages. We didn’t worry about data structures, since we were
simply interested in numbers. We did, however, spend some
effort in separating out modu... | https://ocw.mit.edu/courses/6-001-structure-and-interpretation-of-computer-programs-spring-2005/03e0268a995c2575b1039dd867c8068b_lecture7webhand.pdf |
of a procedure, such as average, without having to
change any of the procedures that use that particular
component. As well, the flow of information between the
modules helps guide us in the creation of the overall set of
procedures.
Thus, when faced with any new computational problem, we want to try to engage in t... | https://ocw.mit.edu/courses/6-001-structure-and-interpretation-of-computer-programs-spring-2005/03e0268a995c2575b1039dd867c8068b_lecture7webhand.pdf |
,
what is the rough description of the process? Note that here we
have been a bit cryptic (in order to fit things on the slide) and
we might well want to say more about “successive refinement”
(though we could defer that to the documentation under the
improve procedure). We also identify the role of each
argument... | https://ocw.mit.edu/courses/6-001-structure-and-interpretation-of-computer-programs-spring-2005/03e0268a995c2575b1039dd867c8068b_lecture7webhand.pdf |
and Interpretation of Computer Programs. Copyright © 2004 by Massachusetts Institute of Technology.
Slide 7.1.16
In general, taking care to meet each of the stages when you
create code will often ensure an easier time when you have to
refine or replace code. Getting into the habit of doing this
every time you writ... | https://ocw.mit.edu/courses/6-001-structure-and-interpretation-of-computer-programs-spring-2005/03e0268a995c2575b1039dd867c8068b_lecture7webhand.pdf |
some procedure. Or, we
may use it as a formal parameter of a procedure, in which case
it gets locally bound to a value when the procedure is applied.
In the last two cases, if we attempt to reference the variable
outside the scope of the binding, that is, somewhere outside the
bounds of the lambda expression in wh... | https://ocw.mit.edu/courses/6-001-structure-and-interpretation-of-computer-programs-spring-2005/03e0268a995c2575b1039dd867c8068b_lecture7webhand.pdf |
we apparently reached the base case of the
if expression, where we hit the unbound variable. We can see in this simple case that our unbound error is
coming from within the body of foo and is in the base case of the decision process.
6.001 Structure and Interpretation of Computer Programs. Copyright © 2004 by Massa... | https://ocw.mit.edu/courses/6-001-structure-and-interpretation-of-computer-programs-spring-2005/03e0268a995c2575b1039dd867c8068b_lecture7webhand.pdf |
of procedure calls before causing an error. Tracking down the original source of this error can be
difficult, as we need to chase our way back through the sequence of expression evaluations to find where we
accidentally created the wrong type of argument.
Slide 7.2.8
The most common sorts of errors, though, are str... | https://ocw.mit.edu/courses/6-001-structure-and-interpretation-of-computer-programs-spring-2005/03e0268a995c2575b1039dd867c8068b_lecture7webhand.pdf |
us to check that
parameters are being updated correctly, and that end cases are
correctly seeking the right termination point. We might
similarly print out the values of intermediate computations within recursive loops, again to ascertain that the
computation is operating with the values we expect, and is computing... | https://ocw.mit.edu/courses/6-001-structure-and-interpretation-of-computer-programs-spring-2005/03e0268a995c2575b1039dd867c8068b_lecture7webhand.pdf |
that fact and small-enuf? already
exist. The basic idea behind this procedure is quite similar to
what we did for square roots. We start with a guess. We then
see how to improve the guess, in this case by computing the
next term in the approximation, which we would like to add in.
If this improvement is small enou... | https://ocw.mit.edu/courses/6-001-structure-and-interpretation-of-computer-programs-spring-2005/03e0268a995c2575b1039dd867c8068b_lecture7webhand.pdf |
. We have gotten better as we are
only computing the odd terms for n, but we are still not right. If
we look again at the mathematical equation, we can see that we
should be alternating signs on each term. Or said another way,
the successive approximations should go up, then down, then
up, then down, and so on. No... | https://ocw.mit.edu/courses/6-001-structure-and-interpretation-of-computer-programs-spring-2005/03e0268a995c2575b1039dd867c8068b_lecture7webhand.pdf |
2004 by Massachusetts Institute of Technology.
Slide 7.3.12
Here is the bug. We started with the wrong initial value – a
common error. By fixing this, we can try again and …
Slide 7.3.13
… finally we get correct performance. Note how we have used
printing of values to isolate changes, as well as using the
debugg... | https://ocw.mit.edu/courses/6-001-structure-and-interpretation-of-computer-programs-spring-2005/03e0268a995c2575b1039dd867c8068b_lecture7webhand.pdf |
a base case value to return (0 in one case,
1 in the other). And there is the idea of applying an operation to
an input value and the result of repeating that process one fewer times. Repeated is intended to capture that
common pattern of operation.
Slide 7.4.3
So here is what we envision: we want our repeated
pro... | https://ocw.mit.edu/courses/6-001-structure-and-interpretation-of-computer-programs-spring-2005/03e0268a995c2575b1039dd867c8068b_lecture7webhand.pdf |
more repetitions to make. In the base case, it needs to do
something, which we have to figure out. And in the recursive
case, we expect to use repeated to solve the smaller
problem of repetition, plus some additional operations, which
we need to figure out.
Slide 7.4.6
For the base case, what do we know?
We know... | https://ocw.mit.edu/courses/6-001-structure-and-interpretation-of-computer-programs-spring-2005/03e0268a995c2575b1039dd867c8068b_lecture7webhand.pdf |
A second way that types can help us is in debugging code. In
particular, we can use the information about types of arguments
and types of return values explicitly to check that procedures
are interacting correctly. And in some cases, where there are
constraints on the actual values being returned, we can also
enfo... | https://ocw.mit.edu/courses/6-001-structure-and-interpretation-of-computer-programs-spring-2005/03e0268a995c2575b1039dd867c8068b_lecture7webhand.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.641 Electromagnetic Fields, Forces, and Motion, Spring 2005
Please use the following citation format:
Markus Zahn, 6.641 Electromagnetic Fields, Forces, and Motion, Spring
2005. (Massachusetts Institute of Technology: MIT OpenCourseWare).
http://ocw.mit.edu (accessed MM DD, ... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/03f46e6626e7e6ac0268ed2e50f6da29_lecture1.pdf |
Lecture 1
Page 1 of 6
3. Gauss’ Law for Electric Field
(cid:118)∫ ε 0E da = ρ dV
∫
i
S
V
≈ 8.854 ×10-12 farads/meter
ε 0 ≈
10-9
36π
1
ε µ 0 0
free space)
c =
3≈
× 108 meters/second (Speed of electromagnetic waves in
4. Gauss’ Law for Magnetic Field
(cid:118)∫ µ 0H da = 0
i
S
In free space:
B = µ H... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/03f46e6626e7e6ac0268ed2e50f6da29_lecture1.pdf |
r
2
Mg
=
r
2l
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 1
Page 3 of 6
3
Mg
⎡2π ε0r
q = ⎢
l
⎣
2
1
⎤
⎥
⎦
III. Faraday Cage
J da = i = -
i
∫(cid:118)
S
d
dt
∫ ρ dV = -
d
dt
(
) =
-q
dq
dt
∫
idt = q
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Za... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/03f46e6626e7e6ac0268ed2e50f6da29_lecture1.pdf |
Tangential H
µ (H - Hbn
an
0
) A = 0
H = Hbn
an
n i ⎡
⎣
H a - H b
⎤ = 0
⎦
(cid:118)
∇ × H = J ⇒ ∫ H i ds = J i da
∫
C
S
H ds - Hatds = Kds
bt
H - Hat = K
bt
n × ⎡
⎣
H a - H b
⎤ = K
⎦
∂ρ
∇ i J + = 0
∂t
5. Conservation of Charge Boundary Condition
i(cid:118)∫
d
∫
J da + ρdV = 0
dt V
S
n i ⎣
⎡ J a -... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/03f46e6626e7e6ac0268ed2e50f6da29_lecture1.pdf |
Kepler’s Second Law
By studying the Danish astronomer Tycho Brahe’s data about the motion of the planets,
Kepler formulated three empirical laws; two of them can be stated as follows:
Second Law A planet moves in a plane, and the radius vector (from the sun to the
planet) sweeps out equal areas in equal times.
Firs... | https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/040b72c08d4f0ef67faf085066e1cc71_MIT18_02SC_MNotes_k.pdf |
ds
;
dt
d
dt
(r × s) =
dr
dt
× s + r ×
ds
.
dt
These rules are just like the product rule for differentiation. Be careful in the second
6 b × a in general.
rule to get the multiplication order correct on the right, since a × b
=
The two rules can be proved by writing everything out in terms of i , j , k c... | https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/040b72c08d4f0ef67faf085066e1cc71_MIT18_02SC_MNotes_k.pdf |
being
central (i.e., directed toward the sun) is equivalent to Kepler’s second law, we need to
translate that law into calculus. “Sweeps out equal areas in equal times” means:
the radius vector sweeps out area at a constant rate .
1
2
KEPLER’S SECOND LAW
The first thing therefore is to obtain a mathematical expr... | https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/040b72c08d4f0ef67faf085066e1cc71_MIT18_02SC_MNotes_k.pdf |
same
plane, and therefore
(4)
r × v has constant direction (perpendicular to the plane of motion).
Since the direction and magnitude of r × v are both constant,
(5)
r × v = K, a constant vector,
and from this we see that
(6)
d
dt
(r × v) = 0 .
But according to the rule (1) for differentiating a vector produc... | https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/040b72c08d4f0ef67faf085066e1cc71_MIT18_02SC_MNotes_k.pdf |
in a plane and area will be swept out by the radius vector at a constant rate.
MIT OpenCourseWare
http://ocw.mit.edu
18.02SC Multivariable Calculus
Fall 2010
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/040b72c08d4f0ef67faf085066e1cc71_MIT18_02SC_MNotes_k.pdf |
55
1.25. The Quantum Group Uq (sl2). Let us consider the Lie algebra
sl2. Recall that there is a basis h, e, f ∈ sl2 such that [h, e] = 2e, [h, f] =
−2f, [e, f] = h. This motivates the following definition.
Definition 1.25.1. Let q ∈ k, q =� ±1. The quantum group U (sl2) is
generated by elements E, F and an invertib... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/041cee4c86b5e5af04423e900bcb36fc_MIT18_769S09_lec06.pdf |
K].
If q is a root of unity, one can also define a finite dimensional version
of Uq (sl2). Namely, assume that the order of q is an odd number �. Let
uq (sl2) be the quotient of Uq (sl2) by the additional relations
E� = F� = K � − 1 = 0.
Then it is easy to show that uq (sl2) is a Hopf algebra (with the co
product i... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/041cee4c86b5e5af04423e900bcb36fc_MIT18_769S09_lec06.pdf |
− q−di
i
, and the q-Serre relations:
(1.26.1)
and
(1.26.2)
1−aij
�
[l]q ![1
i
l=0
l
1)
−
(
a
−
ij
1−aij −lEj Ei
Ei
l = 0, i =�
− l]q !
i
1−aij
�
[l]q ![1
i
l=0
l
1)
−
(
a
−
ij
1−aij −lFj Fl
Fi
i = 0, i =�
− l]q !
i
j
j.
More generally, the same definition can be made for any symmetriz
able Kac-Mood... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/041cee4c86b5e5af04423e900bcb36fc_MIT18_769S09_lec06.pdf |
coalgebra
C, and Primh,g(C) be the space of skew-primitive elements of type
h, g. Then the space Primh,g(H)/k(h − g) is naturally isomorphic to
Ext1(g, h), where g, h are regarded as 1-dimensional right C-comodules.
Proof. Let V be a 2-dimensional H-comodule, such that we have an
exact sequence
0
→ → → →
V
g
... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/041cee4c86b5e5af04423e900bcb36fc_MIT18_769S09_lec06.pdf |
Y = �E, F K�, a 2-dimensional space. The set of iso
morphism classes of nontrivial extensions of K by 1 is therefore the
projective line PY . The operator of conjugation by K acts on Y with
eigenvalues q2, q−2, hence nontrivially on PY . Thus for a generic ex
tension V , the object V ∗∗ is not isomorphic to V .
Ho... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/041cee4c86b5e5af04423e900bcb36fc_MIT18_769S09_lec06.pdf |
�∗ ∼ V (2h∨), where h∨ is the dual
Coxeter number of g, see [CP, p.384]. If V is a non-trivial irreducible
finite dimensional representation then V (z) = V for z = 0. Thus,
V ∗∗ ∼�
= V . Moreover, we see that the functor ∗∗ has infinite order even
when restricted to simple objects of C.
=
∼
However, the representat... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/041cee4c86b5e5af04423e900bcb36fc_MIT18_769S09_lec06.pdf |
(right) action of A on E. Then the matrix of a ∈ A in
the basis v0, v1 has the form
→
[a]1 =
�
ε(a) χ1(a)
ε(a)
�
(1.27.1)
0
where χ1 ∈ A∗ is nonzero. Since a → [a]1 is a homomorphism, χ1 is a
derivation: χ1(xy) = χ1(x)ε(y) + ε(x)χ1(y).
Now consider the representation V ⊗ V . Using the exactness of
tensor pro... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/041cee4c86b5e5af04423e900bcb36fc_MIT18_769S09_lec06.pdf |
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