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...midterm MAE290a (2008) 75 minutes, closed book, no calculators, one page of notes allowed. 1. (20 points) Gaussian elimination. De ne 1 1 0 0 0 0 1 2 1 0 0 0 0 1 2 1 0 0 , A= 0 1 2 1 0 0 0 0 0 1 2 1 0 0 0 0 1 1 1 0 0 b = . 0 0 1 Using the Gaussian elimination procedure discussed in class, nd a solution (if possible) the problem Ax = b. How many solutions are there...
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midterm MAE290a (2008) 75 minutes, closed book, no calculators, one page of notes allowed. 1. (20 points) Gaussian elimination. De ne 1 1 0 0 0 0 1 2 1 0 0 0 0 1 2 1 0 0 , A= 0 1 2 1 0 0 0 0 0 1 2 1 0 0 0 0 1 1 1 0 0 b = . 0 0 1 Using the Gaussian elimination procedure discussed in class, nd a solution (if possible) the problem Ax = b. How many solutions are there to this problem? Is = 0 an eigenvalue of A? Based on the structure of A, what can you say about its eigenvalues and eigenvectors? Construct an LU (or PLU) decomposition of A. 2. (10 points) Existence/uniqueness of the LU decomposition. (a) Is the LU decomposition of A = An n guaranteed to exist? If not, what is/are the condition(s) on A for which the LU decomposition can be proven to exist? (b) If the LU decomposition of A = An n exists, is it uniuqe? If not, what is/are the condition(s) on A for which the LU decomposition can be proven to be unique (given that one exists)? In both cases, describe the conditions on A in terms of both their names and the appropriate short equations. 3. (10 points) Existence/uniqueness of the SVD. Given an arbitrary A = Am n , is the SVD of A guaranteed to exist? Is it unique? Describe clearly how determine to the Moore-Penrose pseudoinverse from the SVD. Is the Moore-Penrose pseudoinverse unique? 4. (20 points) A trace identity for symmetric X. Suppose that X is known to be real and symmetric (that is, xi j = x ji ). Compute trace(AXB) . X Is the result symmetric? Repeat the analysis in the case that X is known to be complex and Hermitian. 5. (30 points) A fourth construction of the SVD. Consider a complete SVD A = U V H of a square matrix A = An n , and write 0 A AH = QDQH 0 where D = diag{ 1 , . . . , n , 1 , . . . , n }. Determine the Q (in terms of U and V ) such that the above formula is satis ed. Extend this formula for the case of tall A = Am n for m > n. Based on this formula, describe (in detail) a fourth construction of the SVD which is an alternative to the three constructions presented in class. In what way(s) is/are this new method superior to the methods based on the eigenvalues of (AAH ) and (AH A)? MAE290a Midterm Homework (due Tuesday, Nov 18): Write a concise yet complete solution set for this exam. Also, - Check your hand derivation of Problem #1 in Matlab using the codes provided in chapter 2 of NR. - Write and test the code to implement the SVD algorithm described in Problem #5 for a full complex matrix A = Am n for m n [hint: leverage [lam,S]=Eig(B, hermitian )].
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UCSD >> MAE >> 290 (Fall, 2008)
MAE290a midterm (2008) 75 minutes, closed book, no calculators, one page of notes allowed. 1. (20 points) Gaussian elimination. Dene 1 1 0 0 0 0 1 2 1 0 0 0 0 1 2 1 0 0 , A= 0 1 2 1 0 0 0 0 0 1 2 1 0 0 0 0 1 1 1 0 0 b = . 0 0 1 Using ...
UCSD >> MAE >> 290 (Fall, 2008)
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UCSD >> MAE >> 99 (Fall, 2008)
j g | g e n g j d n Yv q VqjYe}qVqSewiD| Vj Do~zgwxfjtYVqxjqfgDevrVeg Deph Vqi 8jfgYeYdVd#j Vhpgnv Dqg pYnuY\"VemVq8jfgYeYdVdeg Vhpgn ~n82gt w|Bjti YFvbDrf~eYn y}xuRbrY|rV}kh W8h~gfn6rWjp}w|6jDo#VhpgnWqYh}i b iDtxqRvd8}Vg# DYy}YnYhDqfeWwfgfq~#f...
UCSD >> MAE >> 00 (Fall, 2008)
y gi w 6mxRy@iq}SidxS6Sy}r!xwBmxgBmHgRBiUgzo z nmUgi z}Ugz }Si z BmHgRBiUgzXujdU zdUgR r z|p y jzog y U6|U dSj}6R RmR6 SjR xUknDxgBVgSiUmRrz Ugrz (n g y6m6UxzUwyzr xi h B6mRgRk{lxi}f 6~ dzU6w Bwhg SfRuyUmhUfx rlg UgmxRxunUgHxzRSx}r42...
UCSD >> MAE >> 290 (Fall, 2008)
y gi w 6mxRy@iq}SidxS6Sy}r!xwBmxgBmHgRBiUgzo z nmUgi z}Ugz }Si z BmHgRBiUgzXujdU zdUgR r z|p y jzog y U6|U dSj}6R RmR6 SjR xUknDxgBVgSiUmRrz Ugrz (n g y6m6UxzUwyzr xi h B6mRgRk{lxi}f 6~ dzU6w Bwhg SfRuyUmhUfx rlg UgmxRxunUgHxzRSx}r42...
UCSD >> MAE >> 01 (Spring, 2006)
MAE290a nal (2001) 3 hours, closed book, no calculators, two pages of notes allowed. Neatness counts. 1. Solving nonlinear equations (40 points) 1 A famous Russian mathematician, Alyonka Igorevna, has proposed a new strategy which she thinks might ...
UCSD >> MAE >> 290 (Fall, 2008)
MAE290a nal (2001) 3 hours, closed book, no calculators, two pages of notes allowed. Neatness counts. 1. Solving nonlinear equations (40 points) 1 A famous Russian mathematician, Alyonka Igorevna, has proposed a new strategy which she thinks might ...
UCSD >> MAE >> 02 (Spring, 2006)
1 MAE290a nal (2002) 3 hours, closed book, no calculators, two pages of notes allowed. Neatness counts. Note that the questions/phrases that require answers are indicated with boldface. 1. The Adams-Bashforth method We have seen in class a variety ...
UCSD >> MAE >> 290 (Fall, 2008)
1 MAE290a nal (2002) 3 hours, closed book, no calculators, two pages of notes allowed. Neatness counts. Note that the questions/phrases that require answers are indicated with boldface. 1. The Adams-Bashforth method We have seen in class a variety ...
UCSD >> MAE >> 03 (Spring, 2006)
1 MAE290a nal (2003) 1. The big picture We have discussed in class several methods which we have identied with the following names: A. Bisection B. Brents method C. Conjugate gradient D. Crank-Nicholson E. Cubic spline F. Explicit Euler G. False po...
UCSD >> MAE >> 290 (Fall, 2008)
1 MAE290a nal (2003) 1. The big picture We have discussed in class several methods which we have identied with the following names: A. Bisection B. Brents method C. Conjugate gradient D. Crank-Nicholson E. Cubic spline F. Explicit Euler G. False po...
UCSD >> MAE >> 04 (Spring, 2006)
1 MAE290a nal (2004) 3 hours, closed book, no calculators, two pages of notes allowed. 1. The numerical march of an ODE approximation of a PDE Consider the problem of marching in time a numerical approximation of a simple PDE, the heat equation y 2...
UCSD >> MAE >> 290 (Fall, 2008)
1 MAE290a nal (2004) 3 hours, closed book, no calculators, two pages of notes allowed. 1. The numerical march of an ODE approximation of a PDE Consider the problem of marching in time a numerical approximation of a simple PDE, the heat equation y 2...
UCSD >> MAE >> 05 (Spring, 2006)
1 MAE290a nal (2005) 3 hours, closed book, no calculators, two pages of notes allowed. 1. Adams-Moulton methods We have seen in class a variety of methods to integrate the ordinary differential equation dy = f(y,t) with y(t0 ) = y0 . dt Most of the...
UCSD >> MAE >> 290 (Fall, 2008)
1 MAE290a nal (2005) 3 hours, closed book, no calculators, two pages of notes allowed. 1. Adams-Moulton methods We have seen in class a variety of methods to integrate the ordinary differential equation dy = f(y,t) with y(t0 ) = y0 . dt Most of the...
UCSD >> MAE >> 06 (Spring, 2006)
1 MAE290a nal (2006) 3 hours, closed book, no calculators, two pages of notes allowed. 1. The spurious solution component of the AB2 method Recall from the notes that the AB2 method may be written yn+1 = yn + h 1 f(yn ,tn ) + 2f(yn1 ,tn1 ) . (1) Ap...
UCSD >> MAE >> 290 (Fall, 2008)
1 MAE290a nal (2006) 3 hours, closed book, no calculators, two pages of notes allowed. 1. The spurious solution component of the AB2 method Recall from the notes that the AB2 method may be written yn+1 = yn + h 1 f(yn ,tn ) + 2f(yn1 ,tn1 ) . (1) Ap...
UCSD >> MAE >> 07 (Spring, 2006)
1 MAE290a nal (2007) 3 hours, closed book, no calculators, two pages of notes allowed. 1. Reections and rotations of a vector in Rn Recall the denition H = I wwH where = 2, w= x + e1 , x + e1 = Sign(x1 ) x , and the representative example matrix...
UCSD >> MAE >> 290 (Fall, 2008)
1 MAE290a nal (2007) 3 hours, closed book, no calculators, two pages of notes allowed. 1. Reections and rotations of a vector in Rn Recall the denition H = I wwH where = 2, w= x + e1 , x + e1 = Sign(x1 ) x , and the representative example matrix...
UCSD >> MAE >> 08 (Spring, 2006)
MAE290a nal (2008) Making waves with numerical methods 3 hours, closed book, no calculators, two pages of notes allowed. A small piece of an F18, moving at high velocity, landed in the Lx = 5 m wide Ly = 50 m long swimming pool (which varies in dept...
UCSD >> MAE >> 290 (Fall, 2008)
MAE290a nal (2008) Making waves with numerical methods 3 hours, closed book, no calculators, two pages of notes allowed. A small piece of an F18, moving at high velocity, landed in the Lx = 5 m wide Ly = 50 m long swimming pool (which varies in dept...
UCSD >> MAE >> 290 (Fall, 2008)
1 MAE290a nal solution (1999) 1. Newton Raphson: as the initial guess is to the right of the peak of the function, these iterations will diverge 1. Depending upon how the towards the right, attempting to reach a point to the right of t 6 where f t t...
UCSD >> MAE >> 99 (Fall, 2008)
1 MAE290a nal solution (1999) 1. Newton Raphson: as the initial guess is to the right of the peak of the function, these iterations will diverge 1. Depending upon how the towards the right, attempting to reach a point to the right of t 6 where f t t...
UCSD >> MAE >> 00 (Fall, 2008)
1 MAE290a nal solutions (2000) 1. Computational expense of some simple matrix operations (20 points) (a) Computing MC, where M and C are full: O 2n3 . (b) Computing MC, where M is full and C is diagonal: O n2 . (c) Computing Ax, where A is full: O 2...
UCSD >> MAE >> 290 (Fall, 2008)
1 MAE290a nal solutions (2000) 1. Computational expense of some simple matrix operations (20 points) (a) Computing MC, where M and C are full: O 2n3 . (b) Computing MC, where M is full and C is diagonal: O n2 . (c) Computing Ax, where A is full: O 2...
UCSD >> MAE >> 01 (Spring, 2006)
...
UCSD >> MAE >> 290 (Fall, 2008)
...
UCSD >> MAE >> 02 (Spring, 2006)
1 MAE290a nal solutions (2002) 1. The Adams-Bashforth method # 1 The rst root is the physical one. From this expression, and setting the rst two coefcients in the expansion equal to 1 and 1 2 (the corresponding coefcients in the T...
UCSD >> MAE >> 290 (Fall, 2008)
1 MAE290a nal solutions (2002) 1. The Adams-Bashforth method # 1 The rst root is the physical one. From this expression, and setting the rst two coefcients in the expansion equal to 1 and 1 2 (the corresponding coefcients in the T...
UCSD >> MAE >> 03 (Spring, 2006)
1 MAE290a nal solutions (2003) 1. The big picture Question i) I, V (taking J (x) = 1 xT Ax bT x, choices C, O, U are also appropriate) 2 Question ii) A, O, G (taking J (x) = [ f (x)]2 , choices B, J, L are also acceptable) Question iii) O (taking J...
UCSD >> MAE >> 290 (Fall, 2008)
1 MAE290a nal solutions (2003) 1. The big picture Question i) I, V (taking J (x) = 1 xT Ax bT x, choices C, O, U are also appropriate) 2 Question ii) A, O, G (taking J (x) = [ f (x)]2 , choices B, J, L are also acceptable) Question iii) O (taking J...
UCSD >> MAE >> 04 (Spring, 2006)
1 MAE290a nal solutions (2004) 1. The numerical march of an ODE approximation of a PDE Question 1a: 2 1 1 2 1 1 2 1 A= . (x)2 . 0 Question 1b: Explicit Euler 3 3 0 1 . . 1 , . . 2 1 1 2 Implicit Euler sin(t) 0 10 c(t) = . ....
UCSD >> MAE >> 290 (Fall, 2008)
1 MAE290a nal solutions (2004) 1. The numerical march of an ODE approximation of a PDE Question 1a: 2 1 1 2 1 1 2 1 A= . (x)2 . 0 Question 1b: Explicit Euler 3 3 0 1 . . 1 , . . 2 1 1 2 Implicit Euler sin(t) 0 10 c(t) = . ....
UCSD >> MAE >> 05 (Spring, 2006)
1 MAE290a nal solutions (2005) 1. Adams-Moulton (AM) methods (1A) This implicit method is well suited for linear problems f(y) = Ay for which A has tightly banded sparsity. (1B) (0 h 1)2 + (1 h + 1) + (2h) = 0. (1C) For stability, we need |+ | 1 a...
UCSD >> MAE >> 290 (Fall, 2008)
1 MAE290a nal solutions (2005) 1. Adams-Moulton (AM) methods (1A) This implicit method is well suited for linear problems f(y) = Ay for which A has tightly banded sparsity. (1B) (0 h 1)2 + (1 h + 1) + (2h) = 0. (1C) For stability, we need |+ | 1 a...
UCSD >> MAE >> 06 (Spring, 2006)
1 MAE290a nal solutions (2006) 1. The spurious solution component of the AB2 method Expanding the expression given for leads to 1 1 = (2 )h + (12 + 2)h2 2 + . . . = h h2 2 + . . . 2 2 2 Solving (3a) and (4b) for y0,+ and y0, gives y0,+ = y1 y0 ...
UCSD >> MAE >> 290 (Fall, 2008)
1 MAE290a nal solutions (2006) 1. The spurious solution component of the AB2 method Expanding the expression given for leads to 1 1 = (2 )h + (12 + 2)h2 2 + . . . = h h2 2 + . . . 2 2 2 Solving (3a) and (4b) for y0,+ and y0, gives y0,+ = y1 y0 ...
UCSD >> MAE >> 07 (Spring, 2006)
1 MAE290a nal solutions (2007) 1. Reections and rotations of a vector in Rn (a) H is the Householder reector matrix, and G is the Givens rotation matrix. (b) H H X costs 4n2 ops to compute, and GH X costs 6n ops to compute. 1 0 0 1 (c) H = I 2...
UCSD >> MAE >> 290 (Fall, 2008)
1 MAE290a nal solutions (2007) 1. Reections and rotations of a vector in Rn (a) H is the Householder reector matrix, and G is the Givens rotation matrix. (b) H H X costs 4n2 ops to compute, and GH X costs 6n ops to compute. 1 0 0 1 (c) H = I 2...
UCSD >> MAE >> 08 (Spring, 2006)
MAE290a nal solutions (2008) Making waves with numerical methods 1. Finite diffference approximation of the spatial derivatives The second-order central nite difference approximation of the rst derivative is given by (8.3) in NR: f x = f j+1 f j1 . ...
UCSD >> MAE >> 290 (Fall, 2008)
MAE290a nal solutions (2008) Making waves with numerical methods 1. Finite diffference approximation of the spatial derivatives The second-order central nite difference approximation of the rst derivative is given by (8.3) in NR: f x = f j+1 f j1 . ...
UCSD >> MAE >> 130 (Fall, 2008)
...
UCSD >> MUS >> 2007 (Fall, 2008)
Graduate Key Use / Issuance Policy Once issued a key(s), you are responsible for security of the room and its contents while the room is open during the exercise for which the key(s) is issued. It is your responsibility to return the room to its orig...
UCSD >> MUS >> 2007 (Fall, 2008)
Locker Policy 1. A locker for storage of personal instruments is available to Department of Music Graduate Students, Undergraduate Majors, and Music 32/132 and certain performance ensembles students whose instrument would require a locker. 2. Lockers...
UCSD >> MUS >> 2007 (Fall, 2008)
Classroom A/V Equipment (Media Services) Policies The Department of Music owns a limited inventory of A/V equipment and is available for classroom use. It includes the following: Overhead Projectors DVD/VHS units Video/Data Projectors It is highly...
UCSD >> MUS >> 2007 (Fall, 2008)
Music Department Room Policy The Music Department has classrooms, offices and studio space in 3 buildings: Mandeville Center, Warren Lecture Hall and Humanities and Social Science. The rooms that can be reserved when they are not in use as a classroo...
UCSD >> MUS >> 2007 (Fall, 2008)
Equipment Checkout Guidelines, Responsibilities, Terms of Use Purpose: The Department of Music provides checkout equipment to Department of Music Graduate and Undergraduate Majors to support participation in performance, course work, and other music ...
UCSD >> MUS >> 2007 (Fall, 2008)
Available Checkout Gear Headphones Qmix Headphone Mixer Shure Headphone Amp SA-1 (2) Sony MDR 750 Headphones, for DAT Kits (3) Instruments Bass, Acoustic (w/soft case, no bow) (2) Clarinet, A Clarinet, Alto Clarinet, B flat Clarinet, Bass Clarinet, E...
UCSD >> MUS >> 2007 (Fall, 2008)
Undergraduate Key Use / Issuance Policy September 2007 Once issued a key(s), you are responsible for security of the room and its contents while the room is open during the exercise for which the key(s) is issued. It is your responsibility to return ...
UCSD >> MUS >> 2007 (Fall, 2008)
Practice Room Policy The following is a summary of practice facilities, their availability, and the policies which control their use. Practice Room Use/Scheduling The Department of Music provides key access to practice rooms for the following student...
UCSD >> ECE >> 158 (Winter, 2009)
Homework 1 ECE 158A - Data Networks I Winter 2009 Due: Tuesday January 20th Reading assignment: Please read chapter 1, sections 1.1 - 1.5, and chapter 2, sections 2.1 - 2.6. 1. (5 points) Chapter 1, problem 3, page 70 (3rd edition: problem 3, page...
UCSD >> ECE >> 158 (Winter, 2009)
Solution for HW 1 ECE 158A - Data Networks I 1. Chapter 1, problem 3, page 70. a) We can have n connections between each pair of adjacent switches. This gives a maximum of 4n connections. b) We can have n connections passing through the switch in th...
UCSD >> ECE >> 158 (Winter, 2009)
Homework 2 ECE 158A - Data Networks I - Winter 2009 Due: Thursday January 29th Reading assignment: Please read chapter 3, sections 3.1 - 3.7. 1. (5 points) Design a reliable, pipelined, data transfer protocol that uses only negative ACKs. How quic...
UCSD >> RAMP >> 07 (Fall, 2008)
UNIVERSITY OF CALIFORNIA, SAN DIEGO Detecting Malicious Routers A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Computer Science by Alper Tugay Mzrak Committee in charge: Professor Keit...
UCSD >> RAMP >> 08 (Fall, 2008)
MIZRAK LAYOUT 2/14/08 4:11 PM Page 2 Detecting Compromised Routers via Packet Forwarding Behavior Alper T. Mizrak, VMware Stefan Savage and Keith Marzullo, UCSD Abstract While it is widely understood that criminal miscreants are subverting large ...
UCSD >> RAMP >> 0123 (Fall, 2008)
IEEE TRANSACTIONS ON DEPENDABLE AND SECURE COMPUTING, VOL. 3, NO. 3, JULY-SEPTEMBER 2006 1 Detecting and Isolating Malicious Routers Alper Tugay Mzrak, Student Member, IEEE, Yu-Chung Cheng, Keith Marzullo, Member, IEEE, and Stefan Savage, Member...
UCSD >> RAMP >> 06 (Fall, 2008)
IEEE TRANSACTIONS ON DEPENDABLE AND SECURE COMPUTING, VOL. 3, NO. 3, JULY-SEPTEMBER 2006 1 Detecting and Isolating Malicious Routers Alper Tugay Mzrak, Student Member, IEEE, Yu-Chung Cheng, Keith Marzullo, Member, IEEE, and Stefan Savage, Member...
UCSD >> RAMP >> 05 (Fall, 2008)
Fatih: Detecting and Isolating Malicious Routers Alper Tugay Mzrak, Yu-Chung Cheng, Keith Marzullo and Stefan Savage Dept. of Computer Science and Engineering, University of California, San Diego {amizrak, ycheng, marzullo, savage}@cs.ucsd.edu Abstra...
UCSD >> RAMP >> 05 (Fall, 2008)
W.C. Carter Award The William C. Carter Award is presented annually since 1997 to recognize an individual who has made a significant contribution to the field of dependable computing through his or her graduate dissertation research. The award honor...
UCSD >> RAMP >> 04 (Fall, 2008)
Brief Announcement: Detecting Malicious Routers Alper Tugay Mzrak, Keith Marzullo, Stefan Savage {amizrak, marzullo, savage}@cs.UCSD.edu Categories and Subject Descriptors C.2.2 [Comp.-Comm. Networks]: [Network Protocols] General Terms Reliability,...
UCSD >> RAMP >> 04 (Fall, 2008)
Detecting Malicious Routers Alper T. Mzrak, Keith Marzullo, Stefan Savage University of California, San Diego Introduction Modern packet switched data networks. Data forwarded hop-by-hop from router to router towards destination. Routers can be co...
UCSD >> SOS >> 04 (Fall, 2008)
Fault-Tolerant Forwarding in the Face of Malicious Routers Alper Tugay Mzrak, Keith Marzullo, Stefan Savage University of California, San Diego {amizrak, marzullo, savage}@cs.ucsd.edu We are interested in a simple, yet increasingly important network...
UCSD >> SOS >> 04 (Fall, 2008)
Fault-Tolerant Forwarding in the Face of Malicious Routers Alper Tugay Mzrak, Keith Marzullo, Stefan Savage University of California, San Diego {amizrak, marzullo, savage}@cs.ucsd.edu We are interested in a simple, yet increasingly important network...
UCSD >> SOS >> 04 (Fall, 2008)
Fault-Tolerant Forwarding in the Face of Malicious Routers Alper T. Mzrak, Keith Marzullo, Stefan Savage University of California, San Diego Introduction Modern packet switched data networks. Data must be forwarded hop-by-hop from router to router...
UCSD >> RAMP >> 03 (Fall, 2008)
Structured Superpeers: Leveraging Heterogeneity to Provide Constant-Time Lookup Alper Tugay Mzrak, Yuchung Cheng, Vineet Kumar and Stefan Savage Department of Computer Science & Engineering University of California, San Diego {amizrak, ycheng, vineet...
UCSD >> RAMP >> 03 (Fall, 2008)
Structured Superpeers: Leveraging Heterogeneity to Provide Constant-Time Lookup Alper Mizrak (Presenter) Yuchung Cheng Vineet Kumar Stefan Savage Department of Computer Science & Engineering University of California, San Diego Introduction P2P are...
UCSD >> RAMP >> 01 (Fall, 2008)
Discovering Paths Traversed by Visitors in Web Server Access Logs Alper Tugay Mzrak Department of Computer Engineering Bilkent University 06533 Ankara, TURKEY E-mail: mizrak@cs.bilkent.edu.tr Abstract One of the most difficult and complicated proble...
UCSD >> CS >> 07 (Fall, 2008)
Detecting Compromised Routers via Packet Forwarding Behavior Alper Mizrak Keith Marzullo, and Stefan Savage , University of California, San Diego Abstract While it is widely understood that criminal miscreants are subverting large numbers of Internet...
UCSD >> CS >> 2007 (Fall, 2008)
Detecting Compromised Routers via Packet Forwarding Behavior Alper Mizrak Keith Marzullo, and Stefan Savage , University of California, San Diego Abstract While it is widely understood that criminal miscreants are subverting large numbers of Internet...
UCSD >> TECH >> 07 (Fall, 2008)
Detecting Compromised Routers via Packet Forwarding Behavior Alper Mizrak Keith Marzullo, and Stefan Savage , University of California, San Diego Abstract While it is widely understood that criminal miscreants are subverting large numbers of Internet...
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