Army Ants Tunneling for Classical Simulations
|
|
- Rolf Harper
- 5 years ago
- Views:
Transcription
1 Electronc Supplementary Materal (ESI) for Chemcal Scence. Ths journal s The Royal Socety of Chemstry 2014 electronc supplementary nformaton (ESI) for Chemcal Scence Army Ants Tunnelng for Classcal Smulatons Jngjng Zheng, Xuefe Xu, Rubén Meana-Pañeda, and Donald G. Truhlar* Department of Chemstry, Chemcal Theory Center, and Supercomputng Insttute, Unversty of Mnnesota, Mnneapols, M Corrected: August 19, 2014 Contents Detals of the tunnelng calculaton S-2 Illustraton of the effect of varyng the parameter η S-6
2 S-2 Detals of the tunnelng calculaton Because the tunnelng path s along an nternal coordnate, the dsplacements of Cartesan coordnates along the path are generated by Wlson s A matrx usng eq. 1 Δx = A(x)ΔR (S1) where A(x) s a generalzed nverse matrx of Wlson B matrx at the current geometry x (where x s a vector of 3 Cartesan coordnates, where s the number of atoms), and ΔR s a column vector of nternal coordnate dsplacements. Then the Cartesan coordnates are converted to sonertal coordnates q by eq 2 of the man text. The end of the tunnelng path s the geometry q! at whch # $ V ( q!) V (q 0 )% & becomes zero agan (t s zero at the begnnng of the tunnelng path, wth coordnates q 0, then postve, then comes back to zero). The locaton of the center of mass s unchanged by the tunnelng process because t s carred out n nternal coordnates, and n the presentaton here we place the center of mass at the orgn. Before calculatng a tunnelng path, ts length s unknown. To calculate the magnary acton ntegral effcently n the general case, we predefne a long enough tunnelng path (longer than all the tunnelng paths n the trajectores) and dvde ths predefned path nto segments; n the current studes, the predefned path s 3.6 bohrs for bond length and 180 degrees for torson angle, and the whole path s dvded nto 18 segments. The relatve potental energy of the end pont of each segment relatve to the startng pont of the tunnelng path s calculated, and f the relatve potental energy s postve, the segment should be fully ncluded n the real tunnelng path; f segment M s the frst segment whose end pont has negatve energy, a small step (10-3 bohr for bond
3 S-3 length and 0.1 degree for torson angle) s used to search the precse endng pont of the tunnelng path. To calculate the dstances ξ of ponts from the start of a curved path n sonertal coordnates, an evenly spaced fne grd s created n nternal coordnates along the tunnelng path and the dstance ξ nt n nternal coordnates s calculated for each grd pont. Then ξ n sonertal coordnates s approxmated as a sum of small chord lengths,.e. ξ = q j q j 1. Gauss-Legendre quadrature wth 6 or more nodes s appled to the j=1 whole tunnelng path. For a gven Gauss-Legendre node ξ k that falls between ξ and ξ 1, we use lnear nterpolaton to calculate the correspondng length n nternal nt nt coordnate ξ k,.e., nt (ξ ξk = k ξ 1 )ξ + nt (ξ ξ k )ξ 1 ξ ξ 1. If more than one nternal coordnate s used n the defnton of the tunnelng drecton, ths expresson s used for each nternal coordnate. Once all ξ k nt are known, the Cartesan coordnates of node k are calculated usng Wlson s A matrx teratvely, and then the potental energy s calculated for that Cartesan geometry. To conserve total angular momentum and total energy at the end of the tunnelng path, the fnal atomc momenta are adjusted to satsfy x! p! = x 0, p 0, (S2) =1 =1 p! = p 0, = 1,, (S3) where x 0, and p 0, denote respectvely the ntal poston vector of atom n the unscaled Cartesan coordnates and the ntal momentum of atom. (ote that x 0 n the
4 S-4 man text s a vector of length 3 obtaned by jonng the three components of all x 0, nto a sngle vector.) The prmed varables n eqs. S2 and S3 denote the same quanttes as x 0, and p 0, but at the end of the tunnelng step. The adjustment s accomplshed as follows. The total angular momentum J, whch must be conserved, s J = x p (S4) =1 where x can be x 0, or x! and where p can be p 0, or p!. The change of Cartesan coordnates for atom along the whole tunnelng path s Δx so that x! = x 0, + Δx = 1,, (S5) Equaton S3 conserves the magntudes of the atomc momenta, but not ther drectons. We denote the ntal and fnal atomc momenta as p 0, = p 0, u 0, = p 0, u 0, = 1,, (S6) p! = p 0, u! = p 0, u! = 1,, (S7) where we have used eq. S3, and where u 0, and u! are unt vectors. We choose to mnmze the changes n drecton subject to the constrants of eqs. S2 and S3. Thus we mnmze the quantty f = u 0, " u 2 =1 ( u γ ) 2 =1 γ=x, y,z = u 0,γ " (S8)
5 S-5 subject to the constrant of eq. S2. ote that u 0,γ and u' γ are drecton cosnes. Addng three Lagrange multplers ( λ, =1, 2,3) to enforce the constrant gves a new objectve functon: g = ( u 0,γ u γ # ) 2 & ) + λ ( 1 J x p 0, ( x y # u z # x z # u y # ) + ( + γ=x, y,z ' * & ) & ) +λ ( 2 J y p 0, ( x z # u x # x x # u z # ) + ( + + λ ( 3 J z p 0, ( x x # u y # x y # u x # ) + ( + ' * ' * (S9) Then we combne all the fnal drecton cosnes nto a sngle algebrac vector: v 1 = u 1x,! v 2 = u 1y,! v 3 = u 1z,! v 4 = u 2x,! etc. (S10) Then the equatons to be solved for the fnal drecton cosnes u γ! are g v j = 0, j = 1,, 3 (S11) g λ k = 0. k = 1, 2, 3 (S12) Equatons S11 and S12 consttute nonlnear equatons, and they can be solved teratvely by the ewton-raphson method for the 3 component of v and the three components of λ. Usng the resultng v along wth eqs. S7 and S10, one obtans the momentum components after the tunnelng event.
6 S-6 Illustraton of the effect of varyng the parameter η Fgure S1. Lnearty of the decay curves after nducton tme for varous η values and the lnear fttng results for obtanng rate constant.
Physics 181. Particle Systems
Physcs 181 Partcle Systems Overvew In these notes we dscuss the varables approprate to the descrpton of systems of partcles, ther defntons, ther relatons, and ther conservatons laws. We consder a system
More informationPhysics 5153 Classical Mechanics. Principle of Virtual Work-1
P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal
More informationFinite Element Modelling of truss/cable structures
Pet Schreurs Endhoven Unversty of echnology Department of Mechancal Engneerng Materals echnology November 3, 214 Fnte Element Modellng of truss/cable structures 1 Fnte Element Analyss of prestressed structures
More informationLecture 14: Forces and Stresses
The Nuts and Bolts of Frst-Prncples Smulaton Lecture 14: Forces and Stresses Durham, 6th-13th December 2001 CASTEP Developers Group wth support from the ESF ψ k Network Overvew of Lecture Why bother? Theoretcal
More informationAn Algorithm to Solve the Inverse Kinematics Problem of a Robotic Manipulator Based on Rotation Vectors
An Algorthm to Solve the Inverse Knematcs Problem of a Robotc Manpulator Based on Rotaton Vectors Mohamad Z. Al-az*, Mazn Z. Othman**, and Baker B. Al-Bahr* *AL-Nahran Unversty, Computer Eng. Dep., Baghdad,
More informationSolutions to exam in SF1811 Optimization, Jan 14, 2015
Solutons to exam n SF8 Optmzaton, Jan 4, 25 3 3 O------O -4 \ / \ / The network: \/ where all lnks go from left to rght. /\ / \ / \ 6 O------O -5 2 4.(a) Let x = ( x 3, x 4, x 23, x 24 ) T, where the varable
More informationWeek 9 Chapter 10 Section 1-5
Week 9 Chapter 10 Secton 1-5 Rotaton Rgd Object A rgd object s one that s nondeformable The relatve locatons of all partcles makng up the object reman constant All real objects are deformable to some extent,
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle
More informationModule 3: Element Properties Lecture 1: Natural Coordinates
Module 3: Element Propertes Lecture : Natural Coordnates Natural coordnate system s bascally a local coordnate system whch allows the specfcaton of a pont wthn the element by a set of dmensonless numbers
More informationSection 8.3 Polar Form of Complex Numbers
80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the
More informationProgramming Project 1: Molecular Geometry and Rotational Constants
Programmng Project 1: Molecular Geometry and Rotatonal Constants Center for Computatonal Chemstry Unversty of Georga Athens, Georga 30602 Summer 2012 1 Introducton Ths programmng project s desgned to provde
More informationPhysics 111: Mechanics Lecture 11
Physcs 111: Mechancs Lecture 11 Bn Chen NJIT Physcs Department Textbook Chapter 10: Dynamcs of Rotatonal Moton q 10.1 Torque q 10. Torque and Angular Acceleraton for a Rgd Body q 10.3 Rgd-Body Rotaton
More informationSo far: simple (planar) geometries
Physcs 06 ecture 5 Torque and Angular Momentum as Vectors SJ 7thEd.: Chap. to 3 Rotatonal quanttes as vectors Cross product Torque epressed as a vector Angular momentum defned Angular momentum as a vector
More informationcoordinates. Then, the position vectors are described by
Revewng, what we have dscussed so far: Generalzed coordnates Any number of varables (say, n) suffcent to specfy the confguraton of the system at each nstant to tme (need not be the mnmum number). In general,
More informationIndeterminate pin-jointed frames (trusses)
Indetermnate pn-jonted frames (trusses) Calculaton of member forces usng force method I. Statcal determnacy. The degree of freedom of any truss can be derved as: w= k d a =, where k s the number of all
More informationPhysics 607 Exam 1. ( ) = 1, Γ( z +1) = zγ( z) x n e x2 dx = 1. e x2
Physcs 607 Exam 1 Please be well-organzed, and show all sgnfcant steps clearly n all problems. You are graded on your wor, so please do not just wrte down answers wth no explanaton! Do all your wor on
More information11. Dynamics in Rotating Frames of Reference
Unversty of Rhode Island DgtalCommons@URI Classcal Dynamcs Physcs Course Materals 2015 11. Dynamcs n Rotatng Frames of Reference Gerhard Müller Unversty of Rhode Island, gmuller@ur.edu Creatve Commons
More informationTHE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructions
THE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructons by George Hardgrove Chemstry Department St. Olaf College Northfeld, MN 55057 hardgrov@lars.acc.stolaf.edu Copyrght George
More informationC/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1
C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned
More informationPHYS 705: Classical Mechanics. Calculus of Variations II
1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary
More informationˆ (0.10 m) E ( N m /C ) 36 ˆj ( j C m)
7.. = = 3 = 4 = 5. The electrc feld s constant everywhere between the plates. Ths s ndcated by the electrc feld vectors, whch are all the same length and n the same drecton. 7.5. Model: The dstances to
More informationSupporting Information
Supportng Informaton The neural network f n Eq. 1 s gven by: f x l = ReLU W atom x l + b atom, 2 where ReLU s the element-wse rectfed lnear unt, 21.e., ReLUx = max0, x, W atom R d d s the weght matrx to
More informationAPPENDIX 2 FITTING A STRAIGHT LINE TO OBSERVATIONS
Unversty of Oulu Student Laboratory n Physcs Laboratory Exercses n Physcs 1 1 APPEDIX FITTIG A STRAIGHT LIE TO OBSERVATIOS In the physcal measurements we often make a seres of measurements of the dependent
More informationCHAPTER 10 ROTATIONAL MOTION
CHAPTER 0 ROTATONAL MOTON 0. ANGULAR VELOCTY Consder argd body rotates about a fxed axs through pont O n x-y plane as shown. Any partcle at pont P n ths rgd body rotates n a crcle of radus r about O. The
More information6.3.7 Example with Runga Kutta 4 th order method
6.3.7 Example wth Runga Kutta 4 th order method Agan, as an example, 3 machne, 9 bus system shown n Fg. 6.4 s agan consdered. Intally, the dampng of the generators are neglected (.e. d = 0 for = 1, 2,
More informationNotes on Analytical Dynamics
Notes on Analytcal Dynamcs Jan Peters & Mchael Mstry October 7, 004 Newtonan Mechancs Basc Asssumptons and Newtons Laws Lonely pontmasses wth postve mass Newtons st: Constant velocty v n an nertal frame
More informationInductance Calculation for Conductors of Arbitrary Shape
CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors
More informationPHYS 705: Classical Mechanics. Newtonian Mechanics
1 PHYS 705: Classcal Mechancs Newtonan Mechancs Quck Revew of Newtonan Mechancs Basc Descrpton: -An dealzed pont partcle or a system of pont partcles n an nertal reference frame [Rgd bodes (ch. 5 later)]
More informationTREATMENT OF THE TURNING POINT IN ADK-THEORY INCLUDING NON-ZERO INITIAL MOMENTA
41 Kragujevac J. Sc. 5 (00) 41-46. TREATMENT OF THE TURNING POINT IN ADK-THEORY INCLUDING NON-ZERO INITIAL MOMENTA Vladmr M. Rstć a and Tjana Premovć b a Faculty o Scence, Department o Physcs, Kragujevac
More informationNEWTON S LAWS. These laws only apply when viewed from an inertial coordinate system (unaccelerated system).
EWTO S LAWS Consder two partcles. 1 1. If 1 0 then 0 wth p 1 m1v. 1 1 2. 1.. 3. 11 These laws only apply when vewed from an nertal coordnate system (unaccelerated system). consder a collecton of partcles
More information10. Canonical Transformations Michael Fowler
10. Canoncal Transformatons Mchael Fowler Pont Transformatons It s clear that Lagrange s equatons are correct for any reasonable choce of parameters labelng the system confguraton. Let s call our frst
More informationSpin-rotation coupling of the angularly accelerated rigid body
Spn-rotaton couplng of the angularly accelerated rgd body Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 E-mal: louaelzen@gmal.com November 1, 2017 All Rghts Reserved. Abstract Ths paper s
More informationElectron-Impact Double Ionization of the H 2
I R A P 6(), Dec. 5, pp. 9- Electron-Impact Double Ionzaton of the H olecule Internatonal Scence Press ISSN: 9-59 Electron-Impact Double Ionzaton of the H olecule. S. PINDZOLA AND J. COLGAN Department
More informationCalculus of Variations Basics
Chapter 1 Calculus of Varatons Bascs 1.1 Varaton of a General Functonal In ths chapter, we derve the general formula for the varaton of a functonal of the form J [y 1,y 2,,y n ] F x,y 1,y 2,,y n,y 1,y
More informationFrequency dependence of the permittivity
Frequency dependence of the permttvty February 7, 016 In materals, the delectrc constant and permeablty are actually frequency dependent. Ths does not affect our results for sngle frequency modes, but
More informationMathematical Preparations
1 Introducton Mathematcal Preparatons The theory of relatvty was developed to explan experments whch studed the propagaton of electromagnetc radaton n movng coordnate systems. Wthn expermental error the
More informationCOMPLEX NUMBERS AND QUADRATIC EQUATIONS
COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not
More informationPart C Dynamics and Statics of Rigid Body. Chapter 5 Rotation of a Rigid Body About a Fixed Axis
Part C Dynamcs and Statcs of Rgd Body Chapter 5 Rotaton of a Rgd Body About a Fxed Axs 5.. Rotatonal Varables 5.. Rotaton wth Constant Angular Acceleraton 5.3. Knetc Energy of Rotaton, Rotatonal Inerta
More information6.3.4 Modified Euler s method of integration
6.3.4 Modfed Euler s method of ntegraton Before dscussng the applcaton of Euler s method for solvng the swng equatons, let us frst revew the basc Euler s method of numercal ntegraton. Let the general from
More informationOPTIMISATION. Introduction Single Variable Unconstrained Optimisation Multivariable Unconstrained Optimisation Linear Programming
OPTIMIATION Introducton ngle Varable Unconstraned Optmsaton Multvarable Unconstraned Optmsaton Lnear Programmng Chapter Optmsaton /. Introducton In an engneerng analss, sometmes etremtes, ether mnmum or
More informationSupplemental document
Electronc Supplementary Materal (ESI) for Physcal Chemstry Chemcal Physcs. Ths journal s the Owner Socetes 01 Supplemental document Behnam Nkoobakht School of Chemstry, The Unversty of Sydney, Sydney,
More informationMathematical Modeling to Support Gamma Radiation Angular Distribution Measurements
Mathematcal Modelng to Support Gamma Radaton Angular Dstrbuton Measurements V. Baty, O. Stoyanov Insttute for Safety Problems of Nuclear Power Plants, Natonal Academy of Scences of Ukrane Ukrane D. Fedorchenko,
More informationPhysics for Scientists & Engineers 2
Equpotental Surfaces and Lnes Physcs for Scentsts & Engneers 2 Sprng Semester 2005 Lecture 9 January 25, 2005 Physcs for Scentsts&Engneers 2 1 When an electrc feld s present, the electrc potental has a
More informationMAE140 - Linear Circuits - Fall 10 Midterm, October 28
M140 - Lnear rcuts - Fall 10 Mdterm, October 28 nstructons () Ths exam s open book. You may use whatever wrtten materals you choose, ncludng your class notes and textbook. You may use a hand calculator
More informationElectronic Quantum Monte Carlo Calculations of Energies and Atomic Forces for Diatomic and Polyatomic Molecules
RESERVE HIS SPACE Electronc Quantum Monte Carlo Calculatons of Energes and Atomc Forces for Datomc and Polyatomc Molecules Myung Won Lee 1, Massmo Mella 2, and Andrew M. Rappe 1,* 1 he Maknen heoretcal
More information4DVAR, according to the name, is a four-dimensional variational method.
4D-Varatonal Data Assmlaton (4D-Var) 4DVAR, accordng to the name, s a four-dmensonal varatonal method. 4D-Var s actually a drect generalzaton of 3D-Var to handle observatons that are dstrbuted n tme. The
More informationProfessor Terje Haukaas University of British Columbia, Vancouver The Q4 Element
Professor Terje Haukaas Unversty of Brtsh Columba, ancouver www.nrsk.ubc.ca The Q Element Ths document consders fnte elements that carry load only n ther plane. These elements are sometmes referred to
More informationCOMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD
COMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD Ákos Jósef Lengyel, István Ecsed Assstant Lecturer, Professor of Mechancs, Insttute of Appled Mechancs, Unversty of Mskolc, Mskolc-Egyetemváros,
More informationχ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body
Secton.. Moton.. The Materal Body and Moton hyscal materals n the real world are modeled usng an abstract mathematcal entty called a body. Ths body conssts of an nfnte number of materal partcles. Shown
More informationCollege of Computer & Information Science Fall 2009 Northeastern University 20 October 2009
College of Computer & Informaton Scence Fall 2009 Northeastern Unversty 20 October 2009 CS7880: Algorthmc Power Tools Scrbe: Jan Wen and Laura Poplawsk Lecture Outlne: Prmal-dual schema Network Desgn:
More informationAPPENDIX F A DISPLACEMENT-BASED BEAM ELEMENT WITH SHEAR DEFORMATIONS. Never use a Cubic Function Approximation for a Non-Prismatic Beam
APPENDIX F A DISPACEMENT-BASED BEAM EEMENT WITH SHEAR DEFORMATIONS Never use a Cubc Functon Approxmaton for a Non-Prsmatc Beam F. INTRODUCTION { XE "Shearng Deformatons" }In ths appendx a unque development
More informationIntroduction. - The Second Lyapunov Method. - The First Lyapunov Method
Stablty Analyss A. Khak Sedgh Control Systems Group Faculty of Electrcal and Computer Engneerng K. N. Toos Unversty of Technology February 2009 1 Introducton Stablty s the most promnent characterstc of
More informationConservation of Angular Momentum = "Spin"
Page 1 of 6 Conservaton of Angular Momentum = "Spn" We can assgn a drecton to the angular velocty: drecton of = drecton of axs + rght hand rule (wth rght hand, curl fngers n drecton of rotaton, thumb ponts
More informationGeorgia Tech PHYS 6124 Mathematical Methods of Physics I
Georga Tech PHYS 624 Mathematcal Methods of Physcs I Instructor: Predrag Cvtanovć Fall semester 202 Homework Set #7 due October 30 202 == show all your work for maxmum credt == put labels ttle legends
More informationImplicit Integration Henyey Method
Implct Integraton Henyey Method In realstc stellar evoluton codes nstead of a drect ntegraton usng for example the Runge-Kutta method one employs an teratve mplct technque. Ths s because the structure
More informationModeling of Dynamic Systems
Modelng of Dynamc Systems Ref: Control System Engneerng Norman Nse : Chapters & 3 Chapter objectves : Revew the Laplace transform Learn how to fnd a mathematcal model, called a transfer functon Learn how
More informationMoments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.
Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these
More informationDescription of the Force Method Procedure. Indeterminate Analysis Force Method 1. Force Method con t. Force Method con t
Indeternate Analyss Force Method The force (flexblty) ethod expresses the relatonshps between dsplaceents and forces that exst n a structure. Prary objectve of the force ethod s to deterne the chosen set
More information( ) [ ( k) ( k) ( x) ( ) ( ) ( ) [ ] ξ [ ] [ ] [ ] ( )( ) i ( ) ( )( ) 2! ( ) = ( ) 3 Interpolation. Polynomial Approximation.
3 Interpolaton {( y } Gven:,,,,,, [ ] Fnd: y for some Mn, Ma Polynomal Appromaton Theorem (Weerstrass Appromaton Theorem --- estence ε [ ab] f( P( , then there ests a polynomal
More informationA Hybrid Variational Iteration Method for Blasius Equation
Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 10, Issue 1 (June 2015), pp. 223-229 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) A Hybrd Varatonal Iteraton Method
More informationPhysics 2A Chapter 3 HW Solutions
Phscs A Chapter 3 HW Solutons Chapter 3 Conceptual Queston: 4, 6, 8, Problems: 5,, 8, 7, 3, 44, 46, 69, 70, 73 Q3.4. Reason: (a) C = A+ B onl A and B are n the same drecton. Sze does not matter. (b) C
More informationFormulas for the Determinant
page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use
More informationLagrange Multipliers. A Somewhat Silly Example. Monday, 25 September 2013
Lagrange Multplers Monday, 5 September 013 Sometmes t s convenent to use redundant coordnates, and to effect the varaton of the acton consstent wth the constrants va the method of Lagrange undetermned
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationConvexity preserving interpolation by splines of arbitrary degree
Computer Scence Journal of Moldova, vol.18, no.1(52), 2010 Convexty preservng nterpolaton by splnes of arbtrary degree Igor Verlan Abstract In the present paper an algorthm of C 2 nterpolaton of dscrete
More informationCHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE
CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng
More information3. Be able to derive the chemical equilibrium constants from statistical mechanics.
Lecture #17 1 Lecture 17 Objectves: 1. Notaton of chemcal reactons 2. General equlbrum 3. Be able to derve the chemcal equlbrum constants from statstcal mechancs. 4. Identfy how nondeal behavor can be
More information= = = (a) Use the MATLAB command rref to solve the system. (b) Let A be the coefficient matrix and B be the right-hand side of the system.
Chapter Matlab Exercses Chapter Matlab Exercses. Consder the lnear system of Example n Secton.. x x x y z y y z (a) Use the MATLAB command rref to solve the system. (b) Let A be the coeffcent matrx and
More informationPower law and dimension of the maximum value for belief distribution with the max Deng entropy
Power law and dmenson of the maxmum value for belef dstrbuton wth the max Deng entropy Bngy Kang a, a College of Informaton Engneerng, Northwest A&F Unversty, Yanglng, Shaanx, 712100, Chna. Abstract Deng
More informationKernel Methods and SVMs Extension
Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general
More informationA Fast Computer Aided Design Method for Filters
2017 Asa-Pacfc Engneerng and Technology Conference (APETC 2017) ISBN: 978-1-60595-443-1 A Fast Computer Aded Desgn Method for Flters Gang L ABSTRACT *Ths paper presents a fast computer aded desgn method
More informationLaboratory 1c: Method of Least Squares
Lab 1c, Least Squares Laboratory 1c: Method of Least Squares Introducton Consder the graph of expermental data n Fgure 1. In ths experment x s the ndependent varable and y the dependent varable. Clearly
More informationA NUMERICAL COMPARISON OF LANGRANGE AND KANE S METHODS OF AN ARM SEGMENT
Internatonal Conference Mathematcal and Computatonal ology 0 Internatonal Journal of Modern Physcs: Conference Seres Vol. 9 0 68 75 World Scentfc Publshng Company DOI: 0.4/S009450059 A NUMERICAL COMPARISON
More informationNovember 5, 2002 SE 180: Earthquake Engineering SE 180. Final Project
SE 8 Fnal Project Story Shear Frame u m Gven: u m L L m L L EI ω ω Solve for m Story Bendng Beam u u m L m L Gven: m L L EI ω ω Solve for m 3 3 Story Shear Frame u 3 m 3 Gven: L 3 m m L L L 3 EI ω ω ω
More informationPhysics 141. Lecture 14. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 14, Page 1
Physcs 141. Lecture 14. Frank L. H. Wolfs Department of Physcs and Astronomy, Unversty of Rochester, Lecture 14, Page 1 Physcs 141. Lecture 14. Course Informaton: Lab report # 3. Exam # 2. Mult-Partcle
More informationPhysics 106a, Caltech 11 October, Lecture 4: Constraints, Virtual Work, etc. Constraints
Physcs 106a, Caltech 11 October, 2018 Lecture 4: Constrants, Vrtual Work, etc. Many, f not all, dynamcal problems we want to solve are constraned: not all of the possble 3 coordnates for M partcles (or
More informationSnce h( q^; q) = hq ~ and h( p^ ; p) = hp, one can wrte ~ h hq hp = hq ~hp ~ (7) the uncertanty relaton for an arbtrary state. The states that mnmze t
8.5: Many-body phenomena n condensed matter and atomc physcs Last moded: September, 003 Lecture. Squeezed States In ths lecture we shall contnue the dscusson of coherent states, focusng on ther propertes
More informationLAGRANGIAN MECHANICS
LAGRANGIAN MECHANICS Generalzed Coordnates State of system of N partcles (Newtonan vew): PE, KE, Momentum, L calculated from m, r, ṙ Subscrpt covers: 1) partcles N 2) dmensons 2, 3, etc. PE U r = U x 1,
More informationA Computational Viewpoint on Classical Density Functional Theory
A Computatonal Vewpont on Classcal Densty Functonal Theory Matthew Knepley and Drk Gllespe Computaton Insttute Unversty of Chcago Department of Molecular Bology and Physology Rush Unversty Medcal Center
More informationErratum: A Generalized Path Integral Control Approach to Reinforcement Learning
Journal of Machne Learnng Research 00-9 Submtted /0; Publshed 7/ Erratum: A Generalzed Path Integral Control Approach to Renforcement Learnng Evangelos ATheodorou Jonas Buchl Stefan Schaal Department of
More informationInternational Power, Electronics and Materials Engineering Conference (IPEMEC 2015)
Internatonal Power, Electroncs and Materals Engneerng Conference (IPEMEC 2015) Dynamc Model of Wnd Speed Dstrbuton n Wnd Farm Consderng the Impact of Wnd Drecton and Interference Effects Zhe Dong 1, a,
More informationHomework Notes Week 7
Homework Notes Week 7 Math 4 Sprng 4 #4 (a Complete the proof n example 5 that s an nner product (the Frobenus nner product on M n n (F In the example propertes (a and (d have already been verfed so we
More informationLecture 21: Numerical methods for pricing American type derivatives
Lecture 21: Numercal methods for prcng Amercan type dervatves Xaoguang Wang STAT 598W Aprl 10th, 2014 (STAT 598W) Lecture 21 1 / 26 Outlne 1 Fnte Dfference Method Explct Method Penalty Method (STAT 598W)
More informationSupplement: Proofs and Technical Details for The Solution Path of the Generalized Lasso
Supplement: Proofs and Techncal Detals for The Soluton Path of the Generalzed Lasso Ryan J. Tbshran Jonathan Taylor In ths document we gve supplementary detals to the paper The Soluton Path of the Generalzed
More informationIntroduction to Density Functional Theory. Jeremie Zaffran 2 nd year-msc. (Nanochemistry)
Introducton to Densty Functonal Theory Jereme Zaffran nd year-msc. (anochemstry) A- Hartree appromatons Born- Oppenhemer appromaton H H H e The goal of computatonal chemstry H e??? Let s remnd H e T e
More informationFirst Law: A body at rest remains at rest, a body in motion continues to move at constant velocity, unless acted upon by an external force.
Secton 1. Dynamcs (Newton s Laws of Moton) Two approaches: 1) Gven all the forces actng on a body, predct the subsequent (changes n) moton. 2) Gven the (changes n) moton of a body, nfer what forces act
More informationMMA and GCMMA two methods for nonlinear optimization
MMA and GCMMA two methods for nonlnear optmzaton Krster Svanberg Optmzaton and Systems Theory, KTH, Stockholm, Sweden. krlle@math.kth.se Ths note descrbes the algorthms used n the author s 2007 mplementatons
More informationAGC Introduction
. Introducton AGC 3 The prmary controller response to a load/generaton mbalance results n generaton adjustment so as to mantan load/generaton balance. However, due to droop, t also results n a non-zero
More informationPhysics 53. Rotational Motion 3. Sir, I have found you an argument, but I am not obliged to find you an understanding.
Physcs 53 Rotatonal Moton 3 Sr, I have found you an argument, but I am not oblged to fnd you an understandng. Samuel Johnson Angular momentum Wth respect to rotatonal moton of a body, moment of nerta plays
More informationCHARACTERISTICS OF COMPLEX SEPARATION SCHEMES AND AN ERROR OF SEPARATION PRODUCTS OUTPUT DETERMINATION
Górnctwo Geonżynera Rok 0 Zeszyt / 006 Igor Konstantnovch Mladetskj * Petr Ivanovch Plov * Ekaterna Nkolaevna Kobets * Tasya Igorevna Markova * CHARACTERISTICS OF COMPLEX SEPARATION SCHEMES AND AN ERROR
More informationSome Comments on Accelerating Convergence of Iterative Sequences Using Direct Inversion of the Iterative Subspace (DIIS)
Some Comments on Acceleratng Convergence of Iteratve Sequences Usng Drect Inverson of the Iteratve Subspace (DIIS) C. Davd Sherrll School of Chemstry and Bochemstry Georga Insttute of Technology May 1998
More information( ) 1/ 2. ( P SO2 )( P O2 ) 1/ 2.
Chemstry 360 Dr. Jean M. Standard Problem Set 9 Solutons. The followng chemcal reacton converts sulfur doxde to sulfur troxde. SO ( g) + O ( g) SO 3 ( l). (a.) Wrte the expresson for K eq for ths reacton.
More informationENGN 40 Dynamics and Vibrations Homework # 7 Due: Friday, April 15
NGN 40 ynamcs and Vbratons Homework # 7 ue: Frday, Aprl 15 1. Consder a concal hostng drum used n the mnng ndustry to host a mass up/down. A cable of dameter d has the mass connected at one end and s wound/unwound
More informationLOW BIAS INTEGRATED PATH ESTIMATORS. James M. Calvin
Proceedngs of the 007 Wnter Smulaton Conference S G Henderson, B Bller, M-H Hseh, J Shortle, J D Tew, and R R Barton, eds LOW BIAS INTEGRATED PATH ESTIMATORS James M Calvn Department of Computer Scence
More informationFundamental loop-current method using virtual voltage sources technique for special cases
Fundamental loop-current method usng vrtual voltage sources technque for specal cases George E. Chatzaraks, 1 Marna D. Tortorel 1 and Anastasos D. Tzolas 1 Electrcal and Electroncs Engneerng Departments,
More informationHigh-Order Hamilton s Principle and the Hamilton s Principle of High-Order Lagrangian Function
Commun. Theor. Phys. Bejng, Chna 49 008 pp. 97 30 c Chnese Physcal Socety Vol. 49, No., February 15, 008 Hgh-Orer Hamlton s Prncple an the Hamlton s Prncple of Hgh-Orer Lagrangan Functon ZHAO Hong-Xa an
More informationTransfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system
Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng
More informationLecture Notes on Linear Regression
Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume
More informationEN40: Dynamics and Vibrations. Homework 7: Rigid Body Kinematics
N40: ynamcs and Vbratons Homewor 7: Rgd Body Knematcs School of ngneerng Brown Unversty 1. In the fgure below, bar AB rotates counterclocwse at 4 rad/s. What are the angular veloctes of bars BC and C?.
More informationThe Minimum Universal Cost Flow in an Infeasible Flow Network
Journal of Scences, Islamc Republc of Iran 17(2): 175-180 (2006) Unversty of Tehran, ISSN 1016-1104 http://jscencesutacr The Mnmum Unversal Cost Flow n an Infeasble Flow Network H Saleh Fathabad * M Bagheran
More information