text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
6
6.189 IAP 2007 MIT
Cell Chip
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Michael Perrone © Copyrights by IBM Corp. and by other(s) 2007 17
6.189 IAP 2007 MIT
Cell Features
● Heterogeneous
multicore system
architecture
Power Processor
Element for control
tas... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
2.0.2)
2-Way hardware multithreaded
L1 : 32KB I ; 32KB D
L2 : 512KB
Coherent load / store
VMX-32
Realtime Controls
–
– Software / hardware managed TLB
– Bandwidth / Resource Reservation
– Mediated Interrupts
Locking L2 Cache & TLB
● Element Interconnect Bus (EIB):
Four 16 byte data rings supporting mult... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
controllable using PPE MMIO
DMA 1,2,4,8,16,128 -> 16Kbyte transfers
for I/O access
Two queues for DMA commands: Proxy &
SPU
L
o
c
a
l
S
t
o
r
e
A
U
C
S
P
U
M
F
C
L
o
c
a
l
S
t
o
r
e
A
U
C
S
P
U
M
F
C
N
N
96 Byte/Cycle
N
N
NCU
Power Core
(PPE)
L2 Cache
N
N
MFC
AUC
SPU
Local Store
MFC
AUC
SPU
Local Store ... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
Two configurable interfaces (60GB/s @
5Gbps)
– Configurable number of bytes
– Coherent (BIF) and / or
I/O (IOIFx) protocols
Supports two virtual channels per
interface
Supports multiple system configurations
L
o
c
a
l
S
t
o
r
e
A
U
C
S
P
U
M
F
C
N
L
o
c
a
l
S
t
o
r
e
A
U
C
S
P
U
M
F
C
N
25 GB/sec
XDR DRAM
MIC... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
From Coherent Interconnect
From IOIF0 or IOIF1
–
Interrupt Priority Level Control
Interrupt Generation Ports for IPI
Duplicated for each PPE hardware thread
●
I/O Bus Master Translation (IOT)
Translates Bus Addresses to System
Real Addresses
Two Level Translation
–
I/O Segments (256 MB)
I/O Pages (4K, ... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
AP 2007 MIT
Courtesy of International Business Machines
Corporation. Unauthorized use not permitted.
6.189 IAP 2007
Lecture 2
Cell Performance Characteristics
Michael Perrone © Copyrights by IBM Corp. and by other(s) 2007
23
6.189 IAP 2007 MIT
Why Cell Processor Is So Fast?... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
™ 64 X2
2.4 GHz
Intel
Pentium D®
3.2 GHz
PowerPC®
970MP
2.5 GHz
Cell Broadband
EngineTM
3.2 GHz
Michael Perrone © Copyrights by IBM Corp. and by other(s) 2007
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6.189 IAP 2007 MIT
Cell BE Performance
● BE can outperform a P4/SSE2 at same clock rate by 3 to 18x
(assuming linear scaling) in vario... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
v)
200 fps (IA32)
290 fps (per SPE)
BE Perf
Advantage
8x
8x
2x
6x
12x
15x
14x
10x
6x
18x
12x
12x
Michael Perrone © Copyrights by IBM Corp. and by other(s) 2007
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6.189 IAP 2007 MIT
Michael Perrone © Copyrights by IBM Corp. and by other(s) 2007
1
Key Performance Characteristics
● Cell's performa... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
s by IBM Corp. and by other(s) 2007
29
6.189 IAP 2007 MIT
Cell Application Affinity – Target Industry Sectors
Petroleum Industry
Petroleum Industry
Seismic computing
Seismic computing
Reservoir Modeling,
Reservoir Modeling, ……
Aerospace & Defense
Aerospace & Defense
Signal & Im
Signal & I age Proce... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
2007
Lecture 2
Cell Software Environment
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Cell Software Environment
Programmer
Experience
Development
Environment
Development
Tools Stack
Code Dev Tools
Samples
Workloads
Demos
Debug Tools
SPE Management Lib
Applic... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
s by IBM Corp. and by other(s) 2007
35
6.189 IAP 2007 MIT
Cell Simulator Debugging Environment
Execution Environment
Michael Perrone © Copyrights by IBM Corp. and by other(s) 2007
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6.189 IAP 2007 MIT
Linux on CBE
● Provided as patched to the 2.6.15 PPC64 Kernel
Added heterogeneous lwp/thread model
– SP... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
GNU Libs (glibc,etc)
ILP32 Processes
Library (64-bit)
SPE Object Loader
Services
std. PPC64
elf interp
64-bit GNU Libs (glibc)
LP64 Processes
System Call Interface
exec Loader
File System
Framework
Device
Framework
Network
Framework
Streams
Framework
SPU Management
Framework
Misc format bin
SPU Ob... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
9
6.189 IAP 2007 MIT
Optimized SPE and Multimedia
Extension Libraries
● Standard SPE C library subset
optimized SPE C99 functions including stdlib c lib, math and etc.
subset of POSIX.1 Functions – PPE assisted
Execution Environment
● Audio resample - resampling audio signals
● FFT - 1D and 2D fft functio... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
● VSE_subdiv - variable sharpness subdivision
Execution Environment
algorithm
Michael Perrone © Copyrights by IBM Corp. and by other(s) 2007
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6.189 IAP 2007 MIT
Bringup Workloads / Demos
● Numerous code samples
provided to demonstrate
system design constructs
● Complex workloads and
demos used to evaluat... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
Three modes of debugging SPU threads
– Standalone SPE debugging
– Attach to SPE thread
• Thread ID output when SPU_DEBUG_START=1
Michael Perrone © Copyrights by IBM Corp. and by other(s) 2007
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6.189 IAP 2007 MIT
SPE Performance Tools (executables)
● Static analysis (spu_timing)
Annotates assembly sour... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
: data + code + stack
● Communication
DMA and Bus bandwidth
– DMA granularity – 128 bytes
– DMA bandwidth among LS and System memory
Traffic control
– Exploit computational complexity and data locality to lower data traffic
requirement
Shared memory / Message passing abstraction overhead
Synchronizatio... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
)
● Chassis
Standard IBM BladeCenter form factor with:
7 Blades (for 2 slots each) with full performance
–
2 switches (1Gb Ethernet) with 4 external ports each
–
Updated Management Module Firmware.
External Infiniband Switches with optional FC ports
● Typical Configuration (available today from E&TS)
... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
. IBM may not make these offerings available in
other countries, and the information is subject to change without notice. Consult your local IBM business contact for information on the IBM offerings
available in your area. In no event will IBM be liable for damages arising directly or indirectly from any use of the... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
system dependent and may not be available on Linux. For more information, please
check: http://www.ibm.com/systems/p/software/whitepapers/linux_overview.html
Any performance data contained in this document was determined in a controlled environment. Actual results may vary significantly and are
dependent on many... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
trademarks of Rambus, Inc.
UNIX is a registered trademark in the United States, other countries or both.
Linux is a trademark of Linus Torvalds in the United States, other countries or both.
Fedora is a trademark of Redhat, Inc.
Microsoft, Windows, Windows NT and the Windows logo are trademarks of Microsoft Cor... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
ation, life support, or other hazardous uses where malfunction could result
in death, bodily injury, or catastrophic property damage. The information contained in this document does not affect or change
IBM product specifications or warranties. Nothing in this document shall operate as an express or implied license o... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
-late
● VMX-like SIMD dataflow
Broad set of operations (8 / 16 / 32 Byte)
Graphics SP-Float
IEEE DP-Float
CHANNEL
DMA
SMM
SBI
B
E
B
ATO
RTB
14.5mm2 (90nm SOI)
● Unified register file
128 entry x 128 bit
● 256KB Local Store
Combined I & D
16B/cycle L/S bandwidth
128B/cycle DMA bandwidth
M... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
DMA/DMA-list
Branch hint
● VMX-like SIMD dataflow
Graphics SP-Float
No saturate arith, some byte
IEEE DP-Float (BlueGene-like)
● Unified register file
●
128 entry x 128 bit
256KB Local Store
Combined I & D
16B/cycle L/S bandwidth
128B/cycle DMA bandwidth
● Memory Flow Control (MFC)
LS
L... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
Enqueue DMA Command
Michael Perrone © Copyrights by IBM Corp. and by other(s) 2007
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6.189 IAP 2007 MIT
SPE Block Diagram
Floating-Point Unit
Fixed-Point Unit
Permute Unit
Load-Store Unit
Branch Unit
Channel Unit
Result Forwarding and Staging
Register File
Local Store
(256kB)
Single Port SRAM
I... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
8 PPE-side Command Queue entries
16 SPU-side Command Queue entries
● MMU similar to PowerPC MMU
8 SLBs, 256 TLBs
4K, 64K, 1M, 16M page sizes
Software/HW page table walk
PT/SLB misses interrupt PPE
Atomic Cache Facility
●
4 cache lines for atomic updates
2 cache lines for cast out/MMU reload
● Up to... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
Next Program Counter
SPU Execution Status
SPU Privileged Control
SPU Channel Counter Initialize
SPU Channel Data Initialize
SPU Signal Notification Control
SPU Decrementer Status & Control
MFC DMA Control
MFC Context Save / Restore Registers
SLB Management Registers
4K Physical Page Boundary
4K Physical Page... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
Decrementer Count (Channel 8)
16 Entry MFC Command Queue Interface (Channels 16-21)
DMA Tag Group Query Mask (Channel 22)
Request Tag Status Update (Channel 23)
Immediate
Conditional - ALL
Conditional - ANY
Read DMA Tag Group Status (Channel 24)
DMA List Stall and Notify Tag Status (Channel 25)
DMA List Stall ... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
Data cache region touch (DMA Get hint)
sdcrtst - Data cache region touch for store (DMA Put hint)
sdcrz - Data cache region zero
sdcrs - Data cache region store
sdcrf - Data cache region flush
Synchronization Commands
Lockline (Atomic Update) Commands:
getllar - DMA 128 bytes from EA to LS and set Reservation
p... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
96B/cycle peak bandwidth
Over 100 outstanding requests
Courtesy of International Business Machines Corporation. Unauthorized use not permitted.
Michael Perrone © Copyrights by IBM Corp. and by other(s) 2007
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6.189 IAP 2007 MIT
Element Interconnect Bus – Command Topology
● “Address Concentrator” tree struc... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
Data Arb
16B
16B
16B
16B
16B 16B
16B 16B
16B 16B
16B 16B
MIC
SPE0
SPE2
SPE4
SPE6
BIF/IOIF0
Michael Perrone © Copyrights by IBM Corp. and by other(s) 2007
72
6.189 IAP 2007 MIT
Internal Bandwidth Capability
● Each EIB Bus data port supports 25.6GBytes/sec*
in each direction
● The EIB Command Bus s... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
Controller
Controller
Controller
Controller
Controller
Controller
Controller
Controller
Controller
Controller
Ramp
5Ramp
5
Ramp
7
Ramp
4
Ramp
4
Ramp
8
3
Ramp
Ramp
3
Ramp
9
2
Ramp
Ramp
2
Ramp
10
Ramp
1
Ramp
1
Ramp
11
0
Ramp
Ramp
0
MICMICPPE
SPE0SSPE0PE1
SPE2SSPE2PE3
SPE4SSPE4PE... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
20.110/5.60 Fall 2005
Lecture #5
page
1
Thermochemistry
Much of thermochemistry is based on finding “easy” paths to
calculate changes in enthalpy, i.e. understanding how to work
with thermodynamic cycles.
•
Goal:
To predict H∆ for every reaction, even if it
cannot be carried out in the laboratory
The heat of... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/56de76a110080853b28006a5ce0b0620_l05.pdf |
G. Bawendi, Robert W. Field
20.110/5.60 Fall 2005
Lecture #5
page
2
• ∆
H
f
(° 298.15
K
)
:
The heat of formation is the heat of reaction
to create 1 mole of that compound from its constituent elements in
their most stable forms.
Example (T = 298.15 K)
½ H2 (g,T,1 bar) +... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/56de76a110080853b28006a5ce0b0620_l05.pdf |
•
to thermochemistry]
Reactants
∆ rxH
Products
−
∆∑
ν
i
°
H
, reactants
f i
(
i
)
Elements
°∆∑
H
ν
i
f i
(
, products
i
)
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-2005/56de76a110080853b28006a5ce0b0620_l05.pdf |
H
f
,CH
4
= ∆
°
H
f
,CO
2
∆
H
IV
=
H
2
HO
2
−
−
H H
2
O
2
H
2
= ∆
2
°
H
f
,H O
2
∆
H
rx
= ∆
2
°
H
f
,H O
2
+ ∆
°
H
f
,CO
2
∴
In general,
− ∆
H°
4f
,CH
∆
H
rx
=
∑
ν
i
i
∆
°
H
f i
,
(
products
)
−
∆
°
H
f i
,
(
reactants
)
∑
ν
i
i
ν ≡ stoichiometric coefficient
•
H∆
at constant p and for reversible pV process is
= p... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/56de76a110080853b28006a5ce0b0620_l05.pdf |
calorimeter
adiabatic
constant p
React. (T1) + Cal. (T1)
∆HI
adiabatic
consta
nt p
∆
rxH
→
constant
p
Prod. (T1) + Cal. (T1)
∆
HII
constant
p
Prod. (T2) + Cal. (T2)
I
)
∆ IH
React. (T1) + Cal. (T1)
adiabatic
=
constant p
Prod. (T2) + Cal. (T2)
II)
∆ IIH
Prod. (T2) + Cal. (T2)
=
constant p
Prod. (T1) + Cal.... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/56de76a110080853b28006a5ce0b0620_l05.pdf |
∆
H
II
∴ ∆
H
rx
(
T
1
)
= −
T
2
∫
T
1
+
.
C od Cal dT
Pr
.
(
p
)
•
Constant volume
(when gases involved)
reaction
calorimeter
adiabatic
constant V
R
eact. (T1) + Cal. (T1)
∆UI
adiabatic
consta
nt V
∆
rxU
→
constant V
Prod. (T1) + Cal. (T1)
∆U
II
constant V
Prod. (T2) + Cal. (T2)
I
)
∆ IU
React. (T1) + Cal. (T1)
... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/56de76a110080853b28006a5ce0b0620_l05.pdf |
=
0
0
I
I
(II) Purpose is to m
rom T back to T.
f
2
1
easure heat qV needed to take prod. + cal.
q
V
=
T
∫ 1
T
2
C od C
Pr
.
+
V
(
)
.
al dT
=
∆
U
II
∴ ∆
U
rx
(
T
1
)
= −
T
2
∫
VT
1
(
+
.
C od Cal dT
Pr
.
)
∆
(
)
Now use
H U pV
+
=
or
∆
)
H U pV
+
= ∆
(
Assume only significant contribution to
∆ pV is from gases... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/56de76a110080853b28006a5ce0b0620_l05.pdf |
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 #5
page
7
Temperature dependence of ∆Hrx
uppose know ∆Hrx at some ... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/56de76a110080853b28006a5ce0b0620_l05.pdf |
C prod C react dT
⎣
p
.)
.)
⎤
⎦
−
(
(
p
)
T
2
∫
+ ∆
T
1
CdT
p
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-2005/56de76a110080853b28006a5ce0b0620_l05.pdf |
5.8 The Harmonic Oscillator
To illustrate the beauty and efficiency in describing the dynamics of a quan-
tum system using the dirac notation and operator algebra, we reconsider the
one-dimensional harmonic oscillator discussed in section 4.4.2 and described
by the Hamiltonian operator
with
H =
p2
2m
+
1
2
K x2,
[x, p] =... | https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/575470ae5ba3cafa4fd36e35b0d29e17_harmonic_oscil.pdf |
)
and the Hamiltonian operator can be rewritten in terms of the new operators
a and a+ as
¡
¢
¡
¢
H = ~ω0
2
a+a + aa+
= ~ω0
¡
a+a+
µ
¢
.
1
2
¶
We introduce the operator
N = a+a,
(5.138)
(5.139)
(5.140)
which is a hermitian operator. Up to an additive constant 1/2 and a scaling
factor ~ω0 equal to the energy of one quan... | https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/575470ae5ba3cafa4fd36e35b0d29e17_harmonic_oscil.pdf |
that contains one less quantum
is a new eigenstate to N with eigenvalue Nn
(5.143)
(5.144)
(5.145)
|
|
−
n
i
|
−
¡
|
a
n
i
|
= C
n
|
−
,
1
i
(5.146)
where C is a yet undetermined constant. This constant follows from the
normalization of this state and being an eigenvector to the number operator.
a+a
n
|
h
n
C
=
|
|
i
|... | https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/575470ae5ba3cafa4fd36e35b0d29e17_harmonic_oscil.pdf |
with
i
energy ~ω0/2 we can generate the n-th energy eigenstate by n-fold application
of the creation operator a+ and proper normalization
n + 1
i
n
i
|
= √n + 1
|
|
.
=
n
i
|
1
(n + 1)!
n
a+
,
0
i
|
¡
¢
with
and
p
H
n
i
|
= En
,
n
i
|
En = ~ω0
n+
µ
1
2
.
¶
(5.155)
(5.156)
(5.157)
5.8.2 Matrix Representation
We can expr... | https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/575470ae5ba3cafa4fd36e35b0d29e17_harmonic_oscil.pdf |
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
SLOAN SCHOOL OF MANAGEMENT
15.565 Integrating Information Systems:
Technology, Strategy, and Organizational Factors
15.578 Global Information Systems:
Communications & Connectivity Among Information Systems
Spring 2002
Lecture 9
NETWORK PROTOCOLS
COMPLEXITY OF COMMUNICATION N... | https://ocw.mit.edu/courses/15-565j-integrating-esystems-global-information-systems-spring-2002/57599d12f997dc9b58d783ce08d5353b_lecture09.pdf |
IEEE
5
ISO REFERENCE MODEL FOR OSI
7. APPLICATION: APPLICATION DEPENDENT (E.G., USER PROGRAM)
6. PRESENTATION: COMPRESSION AND CONVERSIONS (E.G., LIBRARY)
5. SESSION: PROCESS-TO-PROCESS (E.G., OS SOFTWARE)
4. TRANSPORT: HOST-TO-HOST (E.G., OS SOFTWARE)
3. NETWORK: ROUTING (E.G., DEVICE DRIVER)
2. DATA: RELIABLE ... | https://ocw.mit.edu/courses/15-565j-integrating-esystems-global-information-systems-spring-2002/57599d12f997dc9b58d783ce08d5353b_lecture09.pdf |
TELEPHONE
• T1 = 1.544M bps (USA & CANADA) OR 2.048M (ELSE)
– SHARED CABLE (ETHERNET)
• 10-100M bps (Typical)
• CSMA/CD (CARRIER SENSE MULTIPLE ACCESS/COLLISION DETECT)
– SATELLITE
• 5-10 CHANNELS, EACH 50M bps
• UP-LINK & DOWN-LINK = 270 MILLISECONDS
• VSAT
– FIBER-OPTIC
• 100M - 10G bps (Typical)
• INTERNE... | https://ocw.mit.edu/courses/15-565j-integrating-esystems-global-information-systems-spring-2002/57599d12f997dc9b58d783ce08d5353b_lecture09.pdf |
NEXT (ACK RECEIVED UP TO)
10
DYNAMIC CHANNEL SHARING
• SATELLITE
– MULTIPLEX (TIME OR FREQUENCY)
– “SLOTTED” ALOHA (PRE-DATES ETHERNET)
LA
• EACH “SLOT”, EITHER SEND OR NO SEND
• IF SEND, LISTEN FOR COLLISION (270 MS)
• IF COLLIDE, TRY AGAIN LATER -- BUT WHEN?
• “BEST ATTAINABLE UTILIZATION” ABOUT 37%
SF
• ... | https://ocw.mit.edu/courses/15-565j-integrating-esystems-global-information-systems-spring-2002/57599d12f997dc9b58d783ce08d5353b_lecture09.pdf |
= COUNTRY NET
10 = NETWORK OPERATOR CHOICE
(E.G., 5 = HOST #, 5 = USER #)
--
--
--
13
SYNCHRONIZATION AND MULTIPLEXING ISSUES
• SYNCHRONIZATION ISSUES:
--
--
--
UNEXPECTED MESSAGE RECEIVED
MULTIPLE PACKETS (DUE TO TIME-OUT & RETRANSMIT)
CLOSING CONNECTIONS
(E.G., TWO ARMY DIVISION PROBLEM:
“YOU ATTACK WHEN YOU... | https://ocw.mit.edu/courses/15-565j-integrating-esystems-global-information-systems-spring-2002/57599d12f997dc9b58d783ce08d5353b_lecture09.pdf |
Transmission Control
Transmission control
Network services
Internet Protocol
Network Access
Path control
Transport
Data link control
Data link control
Physical
Physical
Physical
Approximate correspondences between the various networks.
16
TCP/IP Protocol Architecture
Application layer
Provides communicat... | https://ocw.mit.edu/courses/15-565j-integrating-esystems-global-information-systems-spring-2002/57599d12f997dc9b58d783ce08d5353b_lecture09.pdf |
Lecture 8
8.321 Quantum Theory I, Fall 2017
40
Lecture 8 (Oct. 2, 2017)
8.1 General Time Dependent Hamiltonians
The Schr¨odinger equation dictates that quantum states evolve in time according to
i(cid:126)
d
d
t
| (t)(cid:105) = H(t)|ψ(t)
ψ
(cid:105) .
(8.1)
In the last class, we saw that if the Hamiltonian is independ... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/579cb61c4eaa494b9adb90830d921099_MIT8_321F17_lec8.pdf |
,
U (t, t ) = e−iH(t−t0)/(cid:126)
0
.
(8.2)
(8.3)
(8.4)
(8.5)
(8.6)
(8.7)
2. If [H(t), H(t(cid:48))] = 0 for all t, t(cid:48), then we can simultaneously diagonalize the Hamiltonian at all
times, meaning we can choose a basis of states that are eigenstates of H(t) for all time (the
associated eigenvalues may change as... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/579cb61c4eaa494b9adb90830d921099_MIT8_321F17_lec8.pdf |
We could choose each time interval to have a different length, but the ultimate result will not
be affected by our choice.
We take N large, so that ∆t is small. Then,
(cid:1)
U (ti+1, ti) = 1 − H(ti)∆t + O (∆t)2 .
(cid:0)
i
(cid:126)
If we are ignoring terms of order (∆t)2, then we can write
U (t
i+1
, t ) ≈ e−iH(ti)∆t/(... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/579cb61c4eaa494b9adb90830d921099_MIT8_321F17_lec8.pdf |
48)
(cid:48)
(cid:1)
(cid:19)(cid:21)
.
(8.14)
be
the product in Eq. (8.13) in the
This is called a time-ordered exponential, and is defined to
limit N → ∞, ∆t → 0 with N ∆t held constant. The operator T is called the time-ordering
operator ; it reorders the operators in its argument so that they are time-ordered (disre... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/579cb61c4eaa494b9adb90830d921099_MIT8_321F17_lec8.pdf |
(8.15), we have
U (t, t0) = 1 −
ˆ
t
t0
i
(cid:126)
(cid:1)
dt(cid:48) H t(cid:48) U t(cid:48) t0 .
(cid:0) (cid:1)
(cid:0)
,
(8.17)
This is an expression in terms of the Hamiltonian and
with t(cid:48) ≤ t. We can similarly write
time
-ev
olution operator at times t(cid:48)
U (cid:0)
t(cid:48), t0
(cid:1) = 1 −
ˆ
t(cid:... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/579cb61c4eaa494b9adb90830d921099_MIT8_321F17_lec8.pdf |
1
d 2 · · ·
t
t0
t0
t0
· · ·
dtn H(t1) H(tn) +
· · ·
.
Note that the operators in each integrand are time-ordered. This observation allows us to
write Eq. (8.19) in such a way that the limits of integration are not so complicated, using
ˆ
t(cid:48)
ˆ
t
dt(cid:48)
t0
t
0
dt(cid:48)(cid:48) H
(cid:0)t(cid:48) H
(cid:1) (... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/579cb61c4eaa494b9adb90830d921099_MIT8_321F17_lec8.pdf |
:18)
U (t, t0) = T
exp
−
ˆ
t
t0
i
(cid:126)
dt(cid:48) H t
(cid:0) (cid:48)(cid:1)
(cid:19)(cid:21)
.
(8.22)
8.2
Interaction Picture
We have seen the Schr¨odinger picture and the Heisenberg picture; now we will discuss a third
picture, due to Dirac, called the interaction picture. This is a mixed picture that is useful... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/579cb61c4eaa494b9adb90830d921099_MIT8_321F17_lec8.pdf |
.26)
(8.27)
(8.28)
The interaction picture is useful when we fully understand the dynamics of H0, and V is a weak
probe. We can then use this approach to determine how quickly our system absorbs energy from
the weak probe. In the interaction picture, we are hiding the time evolution due to the system we
understand, so ... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/579cb61c4eaa494b9adb90830d921099_MIT8_321F17_lec8.pdf |
I
dt
= V U = V U0UI ,
i(cid:126)
dU
I
dt
= U −1
0 V U0UI = VI(t)UI(t) .
(8.30)
(8.31)
(8.32)
(8.33)
(8.34)
(8.35)
(8.36)
MIT OpenCourseWare
https://ocw.mit.edu
8.321 Quantum Theory I
Fall 2017
For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/579cb61c4eaa494b9adb90830d921099_MIT8_321F17_lec8.pdf |
(cid:0)(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)
Last
modi(cid:0)ed(cid:1)
September
(cid:2)(cid:3)(cid:4)
(cid:2)(cid:5)(cid:5)(cid:6)
Many(cid:0)body
phenomena
in
condensed
matter
and
atomic
physics
(cid:0)
Lecture
(cid:1)(cid:2)
Vortices(cid:3)
super(cid:4)uidity(cid:2)
Trapped
gases(cid:2)
BEC
at
(cid:5)nite
temperature(... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
(cid:0)(cid:1)(cid:3)
Super(cid:4)uidity(cid:1)
Vortices(cid:1)
Let
us
consider
the
circulation
of
velocity
in
a
super(cid:18)ow(cid:2)
It
follows
from
the
relation
b
e
t
ween
the
velocity
and
the
phase(cid:1)
Eq(cid:2)
(cid:6)(cid:19)(cid:7)(cid:1)
that
the
circulation
around
any
contour
C
obeys
(cid:22) (cid:4)
d
(ci... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
is
the
origin
of
super(cid:18)uidity(cid:2)
The
only
way
to
eliminate
a
super(cid:18)ow
is
to
produce
excitations
with
discrete
vorticity
and
then
remove
them
(cid:6)along
with
the
vorticity(cid:7)
from
the
system(cid:2)
Also
mention
the
Landau
criterion
for
super(cid:18)uidity(cid:25)
The
quasiparticle
energy
(cid:8)
... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
generated
in
the
(cid:18)ow
e
v
en
at
velocities
below
v
(cid:1)
one
cannot
c
use
it
to
predict
the
actual
value
of
critical
velocity(cid:2)
The
observed
critical
velocities
are
system(cid:0)dependent
(cid:6)non(cid:0)universal(cid:7)
and
are
typically
several
orders
of
magntude
below
v
c
estimated
from
quasiparticle
d... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
(cid:11)
(cid:4)
(cid:6)(cid:11)(cid:11)(cid:7)
(cid:4)
p
(cid:1)m
1
For
weakly
interacting
Bose
gas(cid:0)
the
critical
velocity
is
equal
to
Bogoliubov
s
o
u
n
d
v
elocity(cid:1)
(cid:9)
... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
circulation(cid:1)
l
(cid:4)
(cid:11)(cid:1)
we
expect
(cid:0)(cid:6)(cid:9)
(cid:7) (cid:4)
pne
(cid:1)
and
i(cid:0)
(cid:0)(cid:6)(cid:9)
(cid:11) (cid:7)
(cid:9)e
(cid:2)
Thus
one
can
take
a
trial
function
of
the
form
i(cid:0)
(cid:5) (cid:6)
(cid:7)
(cid:8)
(cid:0)(cid:6)
(cid:7) (cid:4)
n
e
(cid:6)(cid:11)(cid:9)(... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
a
vortex
located
on
the
symmetry
axis
of
the
jar
of
radius
b
and
height
L(cid:1)
the
kinetic
energy
of
the
(cid:18)ow
is
L
b
L
b
(cid:1)
(cid:1)
(cid:11)
(cid:8)h
d(cid:9)
(cid:8)h
b
(cid:1)
E
(cid:4)
n(cid:6)(cid:9)(cid:7)
mv
(cid:6)(cid:9)(cid:7)(cid:9)(cid:7)(cid:9)d(cid:9)dz
(cid:4)
(cid:7)
n
dz
(cid:4)
(cid:7)
n
l... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
out
to
b
e
approximately
(cid:7)n
ln
(cid:11)(cid:12)(cid:16)(cid:19)(cid:16)L(cid:1)
which
is
smaller
than
our
m
kinetic
energy
estimate
(cid:6)(cid:11)(cid:14)(cid:7)(cid:2)
The
energy
of
the
vortex
in
a
jar
rotating
with
velocity
(cid:27)
i
s
E
(cid:6)(cid:27)(cid:7)
(cid:4)
E
(cid:27)M
(cid:1)
where
v
v
M
is
the
an... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
:1)
which
means
that
it
is
easier
(cid:6)(cid:3)(cid:7)
to
produce
vortices
in
a
larger
jar(cid:2)
c
(cid:6)(cid:3)(cid:7)
If
the
rotation
velocity
is
larger
than
(cid:27)
and
keeps
increasing(cid:1)
one
can
reach
t
h
e
n
e
x
t
c
critical
value
(cid:27)
at
which
the
second
vortex
appears(cid:1)
and
then(cid:1)
at
some
... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
Of
course(cid:1)
since
N
is
integer(cid:1)
in
reality
the
numb
e
r
of
vortices
increases
discretely(cid:1)
in
steps(cid:1)
on
average
following
the
proportionality
relation
(cid:6)(cid:11)(cid:19)(cid:7)(cid:2)
(cid:0)(cid:1)(cid:5)
Trapped
gases(cid:1)
Bose
condensation
of
con(cid:3)ned
gasses
di(cid:28)ers
somewhat
f... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
to
the
fact
that
the
lowest
energy
quantum
state
in
which
atoms
condense
is
p
e
a
k
ed
at
the
trap
center
and
has
spatial
extent
much
less
than
the
size
of
thermal
cloud
at
temperatures
slightly
above
T
(cid:2)
BEC
In
the
experiments
on
BEC
in
trapped
gasses(cid:1)
atoms
are
con(cid:3)ned
by
a
magnetic
trap(cid:1)
whic... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
e
r
of
atoms
is
large(cid:2)
In
the
presence
of
(cid:5)
(cid:2)
(cid:9)
interactions(cid:1)
on
can
easily
reach
the
limit
when
the
interaction
energy
per
particle
is
much
larger
than
the
level
spacing
in
the
trap(cid:1)
(cid:1)n
(cid:8)h(cid:13) (cid:2)
For
that(cid:1)
the
number
of
atoms
should
exceed
N
(cid:4)
h(cid:... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
)x(cid:7)
dx
(cid:6)(cid:11)(cid:21)(cid:7)
(cid:1)
h
(cid:8)
(cid:11)
(cid:1)
(cid:1)
(cid:8)
(cid:0)
(cid:9)m
jr
j
(cid:0)
j
j
j
j
(cid:1)
(cid:9)
Z
with
the
particle
numb
e
r
N
(cid:4)
(cid:0)
dx
being
(cid:3)xed
by
a
c
hemical
potential
(cid:5)(cid:2)
(cid:1)
Let
us
argue
that
one
can
discard
the
gradient
term
in
t... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
1)N(cid:6)m(cid:13)
(cid:7)
(cid:2)
At
large
(cid:9)
(cid:3)(cid:3)(cid:9)
(cid:1)
2
2
(cid:9)
;
h
(cid:2)
m(cid:2)
(cid:1)
(cid:1)
N
N
(cid:1)
the
value
R
is
much
larger
than
l
that
satis(cid:3)es
l
(cid:4)
l
(cid:2)
Hence
the
c
(cid:2)
(cid:1)m
(cid:1)
(cid:2)
(cid:2)
(cid:10)
2
2
(cid:2)h
m(cid:2)
(cid:1)
(cid:1)
;... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
a
local
density
approximation(cid:1)
i(cid:2)e(cid:2)
treating
each
small
part
of
the
BEC
cloud
as
a
uniform
system(cid:2)
For
the
latter(cid:1)
as
we
already
know(cid:1)
(cid:16)
... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
density
distribution
n(cid:6)x(cid:7)
so
that
the
term
(cid:1)n
in
Eq(cid:2)(cid:6)(cid:11)(cid:23)(cid:7)
compensates
the
potential
U
(cid:6)x(cid:7)
variation
in
space(cid:1)
which
gives
n(cid:6)x(cid:7) (cid:4)
(cid:6)(cid:11)(cid:26)(cid:7)
(cid:0)
(cid:6)(cid:5)
U
(cid:6)x(cid:7)(cid:7)
(cid:6)(cid:1)
(cid:3) U (c... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
3)(cid:3)(cid:9)
R
(cid:4)
N
(cid:6)(cid:9)(cid:9)(cid:7)
(cid:0)
(cid:1)
(cid:16)(cid:7)m
(cid:13)
(cid:1)
For
large
N
N
(cid:1)
the
radius
R
is
much
larger
than
the
BEC
healing
length
(cid:11)
(cid:4)
h(cid:6)
(cid:8)
(cid:1)nm
c
p
estimated
for
typical
density
n
(cid:4)
N(cid:6)
R
(cid:1)
which
determines
the
scale
... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
BEC
coexists
with
the
broad
thermal
distribution(cid:2)
As
temperature
goes
down
and
becomes
very
small(cid:1)
the
thermal
component
in
the
density
distribution
disappears(cid:1)
and
one
obtains
the
zero(cid:0)temperature
state
(cid:6)(cid:9)(cid:15)(cid:7)(cid:2)
(cid:0)(cid:1)(cid:6)
Finite
e(cid:7)ects(cid:8)
quasip... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
Golden
Rule
for
transition
rate(cid:25)
W
(cid:4)
f
i
(cid:16) (cid:6)E
E
(cid:7)
(cid:6)(cid:9)(cid:16)(cid:7)
i
f
int
f
i
!
(cid:9)(cid:7)
(cid:1)
(cid:8)h
jh
jH
j
ij
(cid:0)
f
X
In
a
normal
Bose
gas(cid:1)
at
T (cid:10)
T
(cid:1)
the
rate
of
scattering
out
of
the
state
i
is
BEC
j
i
out
(cid:25)
(cid:4)
M
f
f
(cid:6)... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
Features(cid:25)
(cid:6)(cid:9)(cid:21)(cid:7)
The
rate
vanishes
in
equilibrium(cid:1)
since
(cid:11)
(cid:10)
f
(cid:4)
e
f
(cid:1)
etc(cid:2)
j
j
df
i
(cid:6) (cid:7)
j
(cid:3)
dt
For
near(cid:0)equilibrium
distribution(cid:1)
(cid:4)
(cid:6)f
f
(cid:7)
w
ith
(cid:4)
(cid:7)a
nv
the
classical
dt
(cid:8)
(cid:8)
i
T
i... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
:1)
(cid:4)
M
(cid:6)f
f
(cid:6)(cid:11)
(cid:10)
f
(cid:7)
(cid:6)(cid:11)
(cid:10)
f
(cid:7)(cid:6)(cid:11)
(cid:10)
f
(cid:7)f
(cid:7)
(cid:16) (cid:6)(cid:8)
(cid:10)
(cid:8)
(cid:8)
(cid:7)
(cid:6)(cid:14)(cid:11)(cid:7)
ij(cid:5)m
i
j
m
i
j
m
i
j
m
dt
h(cid:8)
(cid:0)
j
j
(cid:0)
(cid:0)
ij(cid:5)m
X
Due
to
the
p... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
6)(cid:16)(cid:17)(cid:8) c
h
(cid:7)T
T
T
(cid:9)
(cid:9)
(cid:4)
(cid:6)
(cid:2) f
(cid:6)(cid:2) (cid:8)
(cid:7)
(cid:4)
p
p
n
(cid:5)
(cid:5)(cid:3)(cid:1)
(cid:7)
(cid:14)
(cid:0)
Z
(cid:6)(cid:9)(cid:7)(cid:8)
h
(cid:7)
(cid:7)
BEC
(cid:9)
BEC
(cid:9)(cid:6)T
(cid:6)T
(cid:7)
T
T (cid:15)
T
(cid:7)
where
T
(cid:4... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
:1)
a
collective
mode
that
appears
in
the
two(cid:0)(cid:18)uid
regime(cid:2)
In
this
mode(cid:1)
the
relative
fraction
of
the
normal
and
super(cid:18)uid
component
oscillates
and
can
propagate
in
a
sound(cid:0)like
fashion(cid:2)
We
consider
a
uniform
system(cid:1)
in
the
absence
of
external
potential(cid:2)
The
total... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
For
super(cid:18)uid
velocity(cid:1)
o
n
e
can
write
m(cid:2)
(cid:4)
(cid:5)
(cid:6)(cid:14)(cid:17)(cid:7)
v
t
s
(cid:0)r
This
relation(cid:1)
derived
above
from
the
Gross(cid:0)Pitaevskii
equation(cid:1)
is
in
fact
very
general(cid:1)
and
is
true
for
any
super(cid:18)uid(cid:2)
It
follows
from
the
relation
between
t... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
9)
From
Eqs(cid:2)(cid:6)(cid:14)(cid:19)(cid:7)(cid:1)(cid:6)(cid:14)(cid:17)(cid:7)(cid:1)(cid:6)(cid:14)(cid:14)(cid:7)(cid:1)
combined
with
(cid:4)
(cid:9)
(cid:10)
(cid:9)
(cid:1)
obtain
j
v
v
s
s
n
n
(cid:2)
(cid:6)
(cid:7) (cid:4)
(cid:18)
T
(cid:6)(cid:14)(cid:21)(cid:7)
v
v
t
n
s
(cid:9)
(cid:0)
(cid:0)
r
(cid... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
with
(cid:1)
(cid:1)
(cid:1)
(cid:2) p
(cid:9)
(cid:18)
T
(cid:2) p
T
s
(cid:1)
(cid:1)
c
(cid:4)
(cid:3)
c
(cid:4)
(cid:3)
c
(cid:4)
(cid:6)(cid:16)(cid:16)(cid:7)
(cid:3)
(cid:1)
(cid:5)
(cid:0) (cid:1)
(cid:0)
(cid:1)
(cid:9)
C
(cid:9)C
(cid:2)(cid:9)
(cid:2)T
n
T
(cid:10)
(cid:20)
(cid:20)
The
constants
c
and
c
are... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
16)(cid:17)(cid:7)
(cid:0)
(cid:1)
(cid:0)
(cid:1)
(cid:0)
(cid:1)
(cid:2)T
(cid:2)V
(cid:2)(cid:9)
(cid:0)
(cid:10)
T
T
The
I
a
n
d
I
I
sound
velocities(cid:1)
obtained
from
Eq(cid:2)(cid:6)(cid:16)(cid:14)(cid:7)(cid:1)
are
(cid:1)
(cid:1)
(cid:1)
(cid:1)
(cid:11)
(cid:11)
(cid:1)
(cid:1)
(cid:1)
(cid:1)
(cid:1)
(cid... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
ov
quasiparticles
at
low
energies(cid:1)
(cid:9)
(cid:7)
u
(cid:4)
(cid:1)n(cid:6)m(cid:3)
(cid:6)(cid:16)(cid:21)(cid:7)
I
while
the
I
I
sound
velocity
i
s
(cid:14)
times
lower(cid:1)
p
q
u
(cid:4)
u
(cid:6)
(cid:14) (cid:4)
(cid:1)n(cid:6)(cid:14)m(cid:3)
(cid:6)(cid:16)(cid:23)(cid:7)
II
I
p
The
velocity
u
decreases... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
18.409 An Algorithmist’s Toolkit
October 6, 2009
Lecturer: Jonathan Kelner
Scribe: Alessandro Chiesa (2009)
Lecture 8
1 Administrivia
You should probably know that
• the first problem set (due October 15) is posted on the class website, and
• its hints are also posted there.
Also, today in class there was a maj... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/5823d396bfe2ae2c5bea9c431601276e_MIT18_409F09_scribe8.pdf |
in some graph G, we
would like know if v is contained in a cluster, i.e. a subset of vertices that defines a cut with low conductance.
However, we want the running time of our algorithm to depend on the cluster size, and not on the size of
the graph. Last time we mentioned that a good example of a problem of this sor... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/5823d396bfe2ae2c5bea9c431601276e_MIT18_409F09_scribe8.pdf |
that the
random walk is at w; then take the vertices with the k largest probability masses as a possible cut. Repeat
this until you get a good cut or you reach a predetermined limit.
2.3 Obstacles
We need a bound that says that our general strategy works, and that is why we proved the Lov´asz-Simonovits
theorem. H... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/5823d396bfe2ae2c5bea9c431601276e_MIT18_409F09_scribe8.pdf |
time t, if x ≡ w∈W
�
v∈V
dv
�
�
�
�
�
�
w∈W
p t(w) − π(w)� ≤ min
�
�
�
�
�√ √
�
x, 2m − x
�
1
1 − φ(W )2
2
�t
.
Note that in the last lecture we stated a slightly weaker form of the theorem, where the conductance
ϕ(W ) of the cut (W, W ) was replaced by the conductance φ(G) of the whole graph. Ne... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/5823d396bfe2ae2c5bea9c431601276e_MIT18_409F09_scribe8.pdf |
discuss
this approach in the following section.
(Note that for all of this to work we still need to prove a partial converse. Indeed, one can show that if
there exists a cut C of conductance φ2, then at least |C|/2 of its vertices will give a cut of conductance φ,
otherwise the random walk would mix too quickly.)
... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/5823d396bfe2ae2c5bea9c431601276e_MIT18_409F09_scribe8.pdf |
we denote this stationary distribution (which depends on the
parameters s and α) by prα(s) and call it the PageRank vector ; note that prα(s) is a vector in Rn, where
n = |V |. Moreover, it is easy to see that the stationary distribution prα(s) is the unique solution to the
following equation:
prα(s) = αs + (1 − α)... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/5823d396bfe2ae2c5bea9c431601276e_MIT18_409F09_scribe8.pdf |
O( α)).
3.2 Properties
We now prove three properties about the stationary distribution prα.
Proposition 2 (Uniqueness) prα(s) is unique.
Proof We must show that Equation (1) has a unique solution. Rewrite the equation as (I − (1 −
α)W )prα(s) = αs. The matrix I − (1 − α)W is strictly diagonally dominant2 because t... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/5823d396bfe2ae2c5bea9c431601276e_MIT18_409F09_scribe8.pdf |
α)W (cprα(v) + dprα(w))
= αcv + (1 − α)W cprα(v) + αdw + (1 − α)W dprα(w)
= cprα(v) + dprα(w)
(cid:3)
= x .
By Proposition 2, the equation has a unique solution, so that x = x(cid:3) and the result follows.
Proposition 4 (Commutativity with W ) prα(W s) = W prα(s).
Proof By definition, the vector x ≡ prα(W s) sat... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/5823d396bfe2ae2c5bea9c431601276e_MIT18_409F09_scribe8.pdf |
Starting with initial values p = 0 and r = s, in each iteration, we pick a vertex u, and update the two vectors
p and r to the new vectors p(cid:3) and r(cid:3) defined as follows:
p (cid:3) = p + αr(u)χu ,
r (cid:3) = r − r(u)χu + (1 − α)r(u)W χu .
The vector χu is the characteristic vector of u, i.e., the vector wit... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/5823d396bfe2ae2c5bea9c431601276e_MIT18_409F09_scribe8.pdf |
Using the triangle inequality,
||r (cid:3)||1 = ||r − r(u)χu + (1 − α)r(u)W χu||1 ≤ ||r − r(u)χu||1 + (1 − α)r(u)||W χu||1 .
However, ||W χu||1 ≤ 1. Indeed, the ith element of W χu is
1 when i =(cid:8) u and 1 when i = u. Therefore,
2d(u)
2
||r (cid:3)||1 ≤ ||r||1 − r(u) + (1 − α)r(u) = ||r||1 − αr(u) ,
as desired... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/5823d396bfe2ae2c5bea9c431601276e_MIT18_409F09_scribe8.pdf |
9). We deduce that the algorithm must terminate in at most O (cid:4)α
1
Next, by definition, the process terminates when there are no more vertices u such that r(u) ≥ (cid:9)d(u).
iterations.
Therefore, condition (1) is automatically satisfied.
Moreover, if we let T denote the number of iterations that the algorithm ... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/5823d396bfe2ae2c5bea9c431601276e_MIT18_409F09_scribe8.pdf |
C with conductance
φ2, we will find with probability at least 0.5 a set with volume at least vol(C)/2. However, this holds only
if we use “appropriate” parameters α and (cid:9), which we do not know! The fix is to binary search over the
(C) .
α
8-5
possibilities, incurring an additional cost that is only a logarit... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/5823d396bfe2ae2c5bea9c431601276e_MIT18_409F09_scribe8.pdf |
fication
Sparsification is a technique used in dynamic graph algorithms to reduce the dependence of an algorithm’s
time on the number of edges in a graph. We briefly motivate this technique now, and will discuss it next
time.
Suppose that we have a graph G = (V, E) with m = Θ(n2) edges. We would like to solve some cut... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/5823d396bfe2ae2c5bea9c431601276e_MIT18_409F09_scribe8.pdf |
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