MOA19
3 Pages

MOA19

Course Number: C 000821, Fall 2009

College/University: Stanford

Word Count: 1757

Rating:

Document Preview

XX International Linac Conference, Monterey, California LARGE-SCALE SIMULATION OF BEAM DYNAMICS IN HIGH INTENSITY ION LINACS USING PARALLEL SUPERCOMPUTERS Robert D. Ryne and Ji Qiang, LANL, Los Alamos, NM 87545, USA Abstract In this paper we present results of using parallel supercomputers to simulate beam dynamics in next-generation high intensity ion linacs. Our approach uses a three-dimensional space charge...

Unformatted Document Excerpt
Coursehero >> California >> Stanford >> C 000821

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

International XX Linac Conference, Monterey, California LARGE-SCALE SIMULATION OF BEAM DYNAMICS IN HIGH INTENSITY ION LINACS USING PARALLEL SUPERCOMPUTERS Robert D. Ryne and Ji Qiang, LANL, Los Alamos, NM 87545, USA Abstract In this paper we present results of using parallel supercomputers to simulate beam dynamics in next-generation high intensity ion linacs. Our approach uses a three-dimensional space charge calculation with six types of boundary conditions. The simulations use a hybrid approach involving transfer maps to treat externally applied fields (including rf cavities) and parallel particle-in-cell techniques to treat the space-charge fields. The large-scale simulation results presented here represent a three order of magnitude improvement in simulation capability, in terms of problem size and speed of execution, compared with typical twodimensional serial simulations. Specific examples will be presented, including simulation of the spallation neutron source (SNS) linac and the Low Energy Demonstrator Accelerator (LEDA) beam halo experiment. tion can be described by Hamilton's equations, dq H = , dt p dp H =- , dt q (1) where H(q, p, t) denotes the Hamiltonian of the system, and where q and p denote canonical coordinates and momenta, respectively. In the language of mappings we would say that there is a (generally nonlinear) map, M, corresponding to the Hamiltonian H, which maps initial phase space variables, i , into final variables, f , and we write f = M i . (2) 1 INTRODUCTION The high intensity of future accelerator-driven systems places stringent requirements on the allowed beam loss, since very small fractional losses at high energy can produce unacceptably high levels of radioactivity. Previous studies suggest that the low density, large amplitude halo of the beam is a major issue for these systems [1, 2, 3]. Large-scale simulations are an important tool for exploring the beam dynamics, predicting the beam halo, and facilitating design decisions aimed at controlling particle loss and meeting operational requirements. The most widely used model for simulating intense beams in ion rf linacs is represented by the PoissonVlasov equations. These equations are often solved using a particle-in-cell (PIC) approach. In this paper we will describe a parallel simulation capability that combines the PIC method with techniques from magnetic optics, and we will present results of using parallel supercomputers to simulate beam dynamics in high intensity ion rf linacs. The potential in the Hamiltonian includes contributions from both the external fields and the space-charge fields. In the Poisson-Vlasov approach, discreteness effects are neglected and the space charge is represented by a smoothly varying mean field. Typically, the Hamiltonian can be written as a sum of two parts, H = Hext + Hsc, which correspond to the external and space-charge contributions. Such a situation is ideally suited to multi-map symplectic splitoperator methods [4]. A second-order-accurate algorithm for a single step is given by M( ) = M1 ( /2) M2 ( ) M1 ( /2) , (3) where denotes the step size, M1 is the map corresponding to Hext and M2 is the map corresponding to Hsc. This approach can be easily generalized to higher order accuracy using Yoshida's scheme if desired [5]. The electrostatic scalar potential generated by the charged particles is obtained by solving Poisson's equation 2 (r) = -(r)/ 0 . (4) 2 PHYSICAL MODEL AND NUMERICAL METHODS In the PIC approach a number of simulation particles, called macroparticles, are used to solve (indirectly) the evolution equations and model the charged particle dynamics. The motion of individual particles in the absence of radia Work supported by the DOE Grand Challenge in Computational Accelerator Physics, Advanced Computing for 21st Century Accelerator Science and Technology Project, and the Los Alamos Accelerator Code Group using resources at the Advanced Computing Laboratory and the National Energy Research Scientific Computing Center. where is the charge density. We have developed a Fourier-based transformation and an eigenfunction expansion method to handle six different boundary conditions: (1) open in all three dimensions; (2) open transversely and periodic longitudinally; (3,4) round conducting pipe transversely and open or periodic longitudinally; (5,6) rectangular conducting pipe transversely and open or periodic longitudinally. A discussion of the numerical algorithms for solving the Poisson's equation with these different boundary conditions can be found in [8]. The charge density on the grid is obtained by using a volume-weighted linear interpolation scheme[6, 7]. After the potential and electric field is found on the grid, the same scheme is used to interpolate the field at the particle locations. During the course of the simulation each step involves the following: transport of a numerical distribution of particles through a half step based on M1 , solving Poisson's equation based on the particle positions and performing a space-charge "kick" M2 , and performing transport through the remaining half of the step based on M1 . 86 MOA19 XX International Linac Conference, Monterey, California 3 APPLICATIONS We have applied the above 3D parallel PIC approach to an early design of the SNS linac and to the proposed LEDA beam halo experiment. Our simulation of the SNS linac starts at the beginning of the DTL. The code advances particles through drift spaces, quadrupole fields and RF gaps. The dynamics inside the gaps is computed using external fields calculated from the electromagnetic code SUPERFISH [9]. A schematic plot of the SNS linac configuration used in this study is shown in Figure 1 [10]. It consists of three types of RF structures: a DTL, a CCDTL, and a CCL. There are a total of 425 RF segments in the linac. Figure 2 shows the rms transverse size (xrms , yrms ) and the maximum transverse extent (xmax , ymax of ) the bunched beam in the linac with one set of errors. We see that the maximum particle amplitude is well-below the aperture size of the linac. This margin is needed to operate the linac safely and to avoid beam loss at the high energy end. The jump in rms beam size between the DTL and CCDTL at 20 MeV is due to a change of focusing period from 8 to 12 at 805 MHz. are two-fold: first, to study beam halo formation and test our physical understanding of the phenomena, and second, to evaluate our computational models and assess their predictive capability through a comparison of simulation and experiment. Fig. 3 gives a schematic plot of the layout of the experiment [11]. It consists of 52 alternating-focusing quadrupole magnets with a focusing period of 41.96 cm. The gradients of the first four quadrupole magnets can be adjusted to create a mismatch that excites the breathing mode or the quadrupole mode. The transverse beam profile will be measured using a beam-profile scanner. Fig. 4 and Fig. 5 present simulation results of the transverse beam size for the breathing mode and the quadrupole mode, plotted at the center of the drift spaces between quadrupole magnets, as a function of distance. The plots include both the rms beam size and the maximum particle extent in the simulation. The physical parameters for the simulation were I=100 mA, E=6.7 MeV, and f=350 MHz. The simulation was performed using 100 million macroparticles with a 128x128x256 (x-y-z) space-charge grid. Figure 3: LEDA halo experiment layout Figure 1: The SNS linac configuration 2.5 Bore Radius Xrms Yrms Xmax Ymax 2 1 Yrms Xrms Ymax Xmax 0.8 displacement (cm) 0.6 1.5 (cm) 0.4 1 0.2 0.5 0 0 2 4 6 distance (m) 8 10 12 0 0 200 400 600 Kinetic Energy (MeV) 800 1000 Figure 4: Transverse beam size as a function of distance for the breathing mode in the LEDA halo experiment From Fig. 4, the two transverse components of the breathing mode are in phase, while the quadrupole mode in Fig. 5 has the two components out of phase. Evidently, it will be possible in the experiment to clearly excite either of the two modes. Furthermore, the debunching of the beam will not significanly alter the structure of the oscillations. Fig. 6 shows the accumulated one-dimensional 87 Figure 2: Transverse beam size as a function of kinetic energy in the SNS linac In the LEDA beam halo experiment, a mismatched high-intensity proton beam will be propagated through a periodic focusing transport system and measurements will be made of the beam profile. The goals of the experiment MOA19 XX International Linac Conference, Monterey, California 1 Xrms Yrms Xmax Ymax 0.8 Displacement (cm) 0.6 0.4 0.2 0 0 2 4 6 Distance (m) 8 10 12 Figure 5: Transverse beam size as a function of distance for the quadrupole mode in the LEDA halo experiment 800000 Breathing Mode Quadrupole Mode 700000 600000 500000 Figure 7: Horizontal and vertical cumulative density profiles of a quadrupole mode mismatch in the LEDA halo experiment. lations on serial computers are extremely valuable for rapid design and predicting rms properties, large-scale simulations are needed for high-resolution studies aimed at making quantitative predictions of the beam halo. -0.006 -0.004 -0.002 0 x (cm) 0.002 0.004 0.006 0.008 400000 300000 200000 100000 0 -0.008 4 ACKNOWLEDGMENTS Figure 6: Accumulated density profile along x for the breathing mode and the quadrupole mode density profiles (along x) for the breathing mode and the quadrupole mode just after the magnet #49. The breathing mode is more peaked and has a larger extent than the quadrupole mode. Measurements will be taken at this location and will be compared with our simulations. The data in Fig. 6 are well-resolved over a range of about 6 decades. An important piece of information from a design standpoint is amount of charge beyond a specified radius or spatial location as a function of distance along the accelerator. This is shown graphically in Fig. 7 which shows the complement of the horizontal and vertical cumulative density profiles, at every step, when the quadrupole mode is excited. In other words, the contours describe the fraction of charge that would be intercepted by a scraper placed at that transverse position. The above LEDA simulations used 100 million macroparticles and a 3D Poisson solver, and required only 2 hours to execute on 256 processors. In contrast, beam dynamics simulations performed on serial computers typically use 10,000 to 100,000 macroparticles and a 2D Poisson solver. Even if the above large-scale calculations could be performed on a PC, they would require on the order of a month to complete. In conclusion, while small-scale simuMOA19 We thank the SNS linac design team and the LEDA beam halo experiment team for helpful discussions. We also thank S. Habib for helpful discussions and C. Thomas Mottershead for graphics support. 5 REFERENCES [1] R. L. Gluckstern, Phys. Rev. Lett. 73, 1247 (1994). [2] T. P. Wangler, K. R. Crandall, R. Ryne, and T. S. Wang, Phys. Rev. ST Accel. Beams 1, 084201 (1998). [3] J. Qiang and R. D. Ryne, Phys. Rev. ST. Accel. Beams 3, 064201 (2000). [4] E. Forest et al., Phys. Lett. A 158, 99 (1991). [5] H. Yoshida, Phys. Lett. A 150, 262 (1990). [6] R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles, Adam Hilger, New York, 1988. [7] C. K. Birdsall and A. B. Langdon, Plasma Physics Via Computer Simulation, McGraw-Hill Book Company, NY, 1985. [8] J. Qiang and R. Ryne, "High Performance Particle-In-Cell Simulation in a Proton Linac," to be submitted to PRST-AB. [9] J. H. Billen and L. M. Young, "POISSON SUPERFISH", LANL Report LA-UR-96-1834 (revised Jan. 8, 2000). [10] T. Bhatia et al., "Beam Dynamics Design for the 1-GeV 2MW SNS Linac," LANL Report LA-UR-99-3802, 1999. [11] T. Wangler, "LEDA Beam Halo Experiment-Physics and Concept of the experiment," LA-UR-00-3181, 2000. 88

Textbooks related to the document above:

Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

Allan Hancock College - TTIAB - 2007356
Passed by both HousesNew South WalesTow Truck Industry Amendment Bill 2007ContentsPage1 2 3 4 5 Schedule 1 Schedule 2Name of Act Commencement Amendment of Tow Truck Industry Act 1998 No 111 Consequential amendments of other legislation Rep
Allan Hancock College - FMAAAR - 200312003
FINANCIAL MANAGEMENT AND ACCOUNTABILITY AMENDMENT REGULATIONS 2003 (NO. 1) 2003 NO. 105 FINANCIAL MANAGEMENT AND ACCOUNTABILITY AMENDMENT REGULATIONS 2003 (NO. 1) 2003 NO. 105 - TABLE OF PROVISIONS1. Name of Regulations 2. Commencement 3
Stanford - MICU - 1031
Case 2:04-cv-02627-HBDocument 95Filed 02/01/2006Page 1 of 18IN THE UNITED STATES DISTRICT COURT FOR THE EASTERN DISTRICT OF PENNSYLVANIAIN RE VICURON PHARMACEUTICALS, INC. SECURITIES LITIGATIONCIVIL ACTIONThisDocument Relatesto: NO.
Stanford - VRTS - 1026
Stanford - SSTI - 1033
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28Joseph J . Tabasco, Jr . (75484) Nicole Lavallee (165755) Julie J . Bai (227047) BERMAN DeVALERIO PEASE TABACCO BURT & PUCILLO 425 California Street, Suite 210 0 San Francisco,
Stanford - MUSEE - 1029
RIT - SMAM - 314
Computer Assignment 4Due Evening Section 10/22 Day Section 10/231. Consider the observations below on residual flame times in seconds for strips of treated childrens underwear.9.85 9.93 9.75 9.77 9.67 9.87 9.67 9.94 9.85 9.75 9.83 9.92 9.74 9.99
RIT - SMAM - 314
SMAM 314Suggested ProblemsThe solution to these problems will not be collected. Some of them will be gone over in class. The homework quizzes will be on these problems. Also most of the examination questions will be problems similar to these. Cha
RIT - SMAM - 314
SMAM 314 Name_Exam 412 SolutionMark the following statements True (T) or False (F) (10 points) 1. _T_A. The line that best fits points whose X and Y values are negatively correlated should have a negative slope. True because as x increases y decr
RIT - SMAM - 314
SMAM 314Worksheet 5Name_Mark the following statements true or false ( _A. The p value of a test of hypothesis is 0.036. The null hypothesis should be rejected at = .05. _B. The t test is appropriate for testing hypothesis about the mean of a n
RIT - SMAM - 314
SMAM 314Exam 312Name_Mark the following statements true or false (10 points) 1. _F_A. A 95% confidence interval for the mean in a normal population contains the true value of the population mean all of the time. The confidence interval contains
Stanford - WX - 1011
US District Court Civil Docket as of 04/02/2002 Retrieved from the court on Tuesday, September 27, 2005U.S. District Court Western District of Pennsylvania (Pittsburgh)CIVIL DOCKET FOR CASE #: 2:97-cv-00309-DBSSCHWARTZ v. WESTINGHOUSE, et al Ass
Stanford - RYL - 1029
UNITED STATES DISTRICT COURT NORTHERN DISTRICT OF TEXAS DALLAS DIVISIONTDH PARTNERS LLP, Individually and on Behalf of All Others Similarly Situated, Plaintiffs, v. RYLAND GROUP, INC., et al., CIVIL ACTION No. 3:04-CV-0073-B Consolidated with: 3:04
Stanford - OVTI - 1031
Case 3:04-cv-02297-SCDocument 184Filed 07/28/2006Page 1 of 71 MILBERG WEISS BERSHAD & SCHULMAN LLP 2 JEFF S. WESTERMAN (94559) CHERYL A. WILLIAMS (193532) 3 MICHIYO MICHELLE FURUKAWA (234121) One California Plaza 4 300 South Grand Avenue, Sui
Stanford - BRCD - 1034
Case 3:05-cv-02042-CRBDocument 470Filed 10/24/2008Page 1 of 271 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28Jeffrey J. Angelovich (admitted Pro Hac Vice) Bradley E. Beckworth (admitted Pro Hac Vice) Susan Whatley
Utah - MATH - 1080
Utah - MATH - 2201
Utah - MATH - 2201
Utah - MATH - 1080
MATH 1080, SPRING 2006, PRACTICE EXAM 3 1. Go over all HW problems in HW 8,9, and 10 with solution key posted on the website. 2. Memorize all denitions and formula/rules. Also go over the examples related to the graphs with area to understand the rel
Utah - MATH - 1080
MATH 1080, SPRING 2006, HW SET 2 You need to show all your work and explain with following the guideline described in the class webpage to get the full credit. HW 2 Due on Thursday 01/26/06 0. Memorize all definitions and go over examples given in cl
Utah - MATH - 1080
MATH 1080, SPRING 2006, HW SET 8 You need to show all your work and explain with following the guideline described in the class webpage to get the full credit. And please staple your HW papers. HW 7 Due on Thursday 03/30/06 Before doing this HW, go o
Utah - MATH - 1080
MATH 1080, SPRING 2006, PRACTICE FINAL 1. Go over Exam 1, Exam 2, and Exam 3 problems as well as HW 1-HW 10 for the old materials. 2. For the recent materials(Volume and integration by substitution), study the following problems and related HW proble
Utah - LAW - 113
Utah - LAW - 112
Utah - MATH - 1170
Tentative schedule for MATH 1170, Fall 2007 We will deviate from this schedule throughout the semester, but this gives a good impression of our general focus. (Lecture days are numbered. Section numbers correspond to Adler book) 1. Aug 20: Variables,
Stanford - ADEX - 1036
Case 1:06-cv-11033-PBSDocument 25-1Filed 06/27/2006Page 1 of 24UNITED STATES DISTRICT COURT DISTRICT OF MASSACHUSETTSDEAN DRULIAS, on behalf of himself and all others similarly situated, Plaintiff, Civil Action No. 06-11033 PBS v. ADE CORPO
Stanford - WFC - 1035
1 2 3 4 5 6 7 8 9 10 11 12 13PILLSBURY WINTHROP SHAW PITTMAN LLP BRUCE A. ERICSON #76342 CLIFFORD C. HYATT #196458 DAVID L. STANTON #208079 JACOB R. SORENSEN #209134 RYAN TAKEMOTO #221169 50 Fremont Street Post Office Box 7880 San Francisco, CA 941
Stanford - WFC - 1035
1 2 3 4 5 6 7 8 9 10 11 12 13PILLSBURY WINTHROP SHAW PITTMAN LLP BRUCE A. ERICSON #76342 CLIFFORD C. HYATT #196458 DAVID L. STANTON #208079 JACOB R. SORENSEN #209134 RYAN TAKEMOTO #221169 50 Fremont Street Post Office Box 7880 San Francisco, CA 941
Stanford - SLF - 1029
IN THE UNITED STATES DISTRICT COURT FOR THE DISTRICT OF MARYLANDIN RE MUTUAL FUNDS INVESTMENT LITIGATIONMDL 1586 Case No. 04-MD-15863 (Judge J. Frederick Motz)This Document Relates To: In re MFS 04-md-15863-04BRUCE RIGGS, et al., Individually
Stanford - SIR - 1033
Case 1:04-cv-07644Document 39Filed 10/19/2005Page 1 of 162IN THE UNITED STATES DISTRICT COURT FOR THE NORTHERN DISTRICT OF ILLINOIS EASTERN DIVISION CENTRAL LABORERS' PENSION FUND, ) No. 04 C-7644 ) Plaintiff, ) Judge Ronald A. Guzman ) v. )
Stanford - XXFFAPK - 1033
Case 3:05-cv-00969-MMCDocument 195Filed 09/14/2006Page 1 of 31 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28DARRYL P. RAINS (CA SBN 104802) GRACE Y. PARK (CA SBN 239928) MORRISON & FOERSTER LLP 755 Page Mill Road
Stanford - XXFFAPK - 1033
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28DARRYL P . RAINS (CA SBN 104802) GRACE Y . PARK (CA SBN 239928) MORRISON & FOERSTER LL P 755 Page Mill Road Palo Alto, California 943 04-1 0 1 8 Telephone : 650 .813 .5600 Fa
Stanford - VECO - 1033
UNITED STATES DISTRICT COURT SOUTHERN DISTRICT OF NEW YORK L.I.S.T., INC., On Behalf of Itself and All Others Similarly Situated, ) ) ) Plaintiff, ) ) vs. ) ) VEECO INSTRUMENTS, INC., EDWARD H. ) BRAUN and JOHN F. REIN, JR., ) ) Defendants. ) Civ. N
San Jose State - CS - 147
. Draw the truth table of the following circuit. The inputs are X and Y, and the outputs are F and G.X YF GX 0 0 1 1Y 0 1 0 1F 1 0 0 0G 0 1 0 1Suppose that the propagation of each gate is 1ns. If the inputs change at time = 0, at what t
Allan Hancock College - AMASOAOEB - 2006788
Serial 35 Assembly Members and Statutory Officers (Remuneration and Other Entitlements) Bill 2006 Ms MartinA BILL for AN ACTto provide for the remuneration and other entitlements of Assembly members and statutory officers, and for related purposes
San Jose State - ME - 120
Field Wiring and Noise Considerations for Analog Signals - Tutorial - D.http:/zone.ni.com/devzone/conceptd.nsf/webmain/01F147E156A1BE1.view cart | help | search NI Developer Zone6Yo u a r e h e r e : N I H o m e > N I D e v e l o p e r Z o n
San Jose State - CS - 165
S curity Engine ring e eC hapte 6 r Distribute S m d yste sS curity Engine ring (Chapte 6) e e r1I ntro S curity Engine ring: A Guideto Building e eDe ndableDistribute S m by Ross pe d yste s, Ande rson C hapte 6: Distribute S m r d yste s
Allan Hancock College - HAB - 2004159
Western Australia Health Amendment Bill 2004 CONTENTS1. Short title 12. Commencement 23. The Act amended
RIT - P - 09233
High Precision Tri-Axis Inertial Sensor ADIS16350/ADIS16355FEATURESTri-axis gyroscope with digital range scaling 75/s, 150/s, 300/s settings 14-bit resolution Tri-axis accelerometer 10 g measurement range 14-bit resolution 350 Hz bandwidth Factory
Stanford - SEP - 110
Stanford Exploration Project, Report 110, September 18, 2001, pages 175Short Note On asymmetric Stolt residual migration for imaging under salt edgesDaniel Rosales and Biondo Biondi 1INTRODUCTION Imaging under salt remains a problem for the oil
Stanford - SEP - 112
Stanford Exploration Project, Report 112, November 11, 2002, pages 2127Short Note Wave-equation MVA using diffracted dataPaul Sava and John Etgen1INTRODUCTION Migration velocity analysis (MVA) using diffracted data is not a new concept. Harlan (
Stanford - MEETING - 060228
Residual z0bestZ0Res = aZpdTrackDigi->z0() extr.z0() ; Extr => ExtrapolationBeginning Run 5bLatest Run 5bBeginning Run 5bLatest Run 5b
Utah - EE - 3110
Utah - ME - 4050
CONCURRENT ENGINEERING ME 4050 Homework 6 Problems 3.1, 3.3, 3.6, 3.8, 3.10Problem 3.1A given type of pressure switch has an estimated failure rate of 24 x 10^-6 failures per cycle. What is the reliability?f := 24 106R := 1 fR = 0.999976
Stanford - C - 070910
MENU 2007 11th International Conference on Meson-Nucleon Physics and the Structure of the Nucleon September10-14, 2007 IKP, Forschungzentrum Jlich, GermanySEARCH FOR -NUCLEUS BOUND STATESV. Jha ,1 , A. Budzanowski% , A. Chatterjee , P. Hawranek ,
Stanford - C - 011127
8th International Conference on Accelerator & Large Experimental Physics Control Systems, 2001, San Jose, CaliforniaWECT002 physics/0111104THE WAVEFRONT CONTROL SYSTEM FOR THE NATIONAL IGNITION FACILITYLewis Van Atta, Mark Perez, Richard Zachari
Allan Hancock College - PSMEAAB - 2004597
House of Assembly No 27As laid on the table and read a first time, 13 October 2004South AustraliaPublic Sector Management (Chief Executive Accountability) Amendment Bill 2004A Bill ForAn Act to amend the Public Sector Management Act 1995.
Stanford - CS - 205
CS205 Homework #2 SolutionsProblem 1[Heath 3.29, page 152] Let v be a nonzero n-vector. The hyperplane normal to v is the (n-1)-dimensional subspace of all vectors z such that vT z = 0. A reflector is a linear transformation R such that Rx = -x if
Stanford - BRCD - 1034
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28Laurence D. King (State Bar No. 206423) KAPLAN FOX & KILSHEIMER LLP 555 Montgomery Street, Suite 1501 San Francisco, CA 94111 Telephone: 415-772-4700 Facsimile: 415-772-4707
Stanford - GEXCOB - 1031
-'"'" "07/15/2004 , -,16:07 FAX 713 227 9404-.-'.HOEFFNER BILEK" &"~005/0281J, ,." '"~"':'r, .;,~. '.' 't ,". "':'ot'"" I,".'" i.UNITEDSTATES COURTSSOUTHERNDISTRICT TEXAS OFUNITED STATES DISTRICT COURT SOUTHERN DISTRICT
Stanford - REI - 1024
UNITED STATES DISTRICT COURT SOUTHERN DISTRICT OF TEXAS HOUSTON DIVISION IN RE RELIANT SECURITIES LITIGATION THIS DOCUMENT RELATES TO: ALL ACTIONS ) ) ) ) ) ) ) CIVIL ACTION NO. H-02-1810 (CONSOLIDATED) Judge Ewing Werlein, Jr.NOTICE OF PENDENCY OF
Stanford - REI - 1024
a./UNITED STATES DISTRICT COURT SOUTHERN DISTRICT OF TEXAS HOUSTON DIVISIO NU'0-~ 0 PYIN RE RELIANT SECURITIES LITIGATION: No. H-02-1810 (CONSOLIDATED) Judge Ewing Werlein, Jr .THIS DOCUMENT RELATES TO : ALL ACTION SCONSOLIDATED CLASS ACT
Utah - MATH - 32103
Math 3210-3 HW 3Due Friday, August 31, 2007There are 7 points possible on this assignment. Only part (b) of problem 3 will be graded.Set Operations1. Prove: (A r B) (B r A) = (A B) r (A B). 2. Prove: A B = A r (A r B). 3. Let {Aj : j J} b
Utah - MATH - 1030
Math 1030Practice Final ExamSpring 2007Name:Instructions: This test is out of 100 points. Please show your work on each question and place your answer in the space provided.The formulas below are provided for your convenienceSavings Plan F
Utah - U - 0387793
Word 2002 (aka XP) - Part 2 (Intermediates)Introduction This guide is presented from the PC perspective. Word 2003 for Macs works similarly. This guide merely introduces the capabilities of Word 2002. Explore, and learn more. Troubleshooting &
Stanford - ADPI - 1039
Case 1:08-cv-10119-RGSDocument 6Filed 03/31/2008Page 1 of 3UNITED STATES DISTRICT COURT DISTRICT OF MASSACHUSETTS MICHAEL OLIPHANT, Individually and On Behalf of All Others Similarly Situated, Plaintiff, v. GREGORY A. SERRAO, BREHT T. FEIGH,
Stanford - NATIONALPA - 1012
Utah - LAW - 138
Stanford - CSCO - 1018
Case 5:01-cv-20418-JWDocument 627Filed 11/28/2006Page 1 of 171 2 3 4 5 6 7 8 9 10101 California Street San Francisco, CA 94111-5894DEAN S. KRISTY (State Bar No. 157646) KEVIN P. MUCK (State Bar No. 120918) FELIX S. LEE (State Bar No. 19708
Utah - LECTURE - 6603
Lecture 15Boundary Layer Theory & Analogies between Mass, Momentum, and Heat Transfer1.0 What is Boundary Layer Theory?1.1 Based on observation that for many problems the transport of momentum, energy, and mass takes place in regions near interf
Allan Hancock College - AB - 2000285
1 Appropriation (Parliament)APPROPRIATION (PARLIAMENT) BILL 2000 EXPLANATORY NOTESGENERAL OUTLINEPolicy Objectives of the Bill This Bill p