SCALAR A physical quantity that is completely characterized by a real number (or by its numerical value) is called a scalar. In other words, a scalar

Size: px
Start display at page:

Download "SCALAR A physical quantity that is completely characterized by a real number (or by its numerical value) is called a scalar. In other words, a scalar"

Transcription

1

2 SCALAR A phscal quantt that s completel charactered b a real number (or b ts numercal value) s called a scalar. In other words, a scalar possesses onl a magntude. Mass, denst, volume, temperature, tme, energ, area, speed and length are eamples to scalar quanttes.

3 ECTOR Several quanttes that occur n mechancs requre a descrpton n terms of ther drecton as well as the numercal value of ther magntude. Such quanttes behave as vectors. Therefore, vectors possess both magntude and drecton; and the obe the parallelogram law of addton. Force, moment, dsplacement, veloct, acceleraton, mpulse and momentum are vector quanttes.

4 Tpes of ectors Phscal quanttes that are vectors fall nto one of the three classfcatons as free, sldng or fed. A free vector s one whose acton s not confned to or assocated wth a unque lne n space. For eample f a bod s n translatonal moton, veloct of an pont n the bod ma be taen as a vector and ths vector wll descrbe equall well the veloct of ever pont n the bod. Hence, we ma represent the veloct of such a bod b a free vector. In statcs, couple moment s a free vector.

5 A sldng vector s one for whch a unque lne n space must be mantaned along whch the quantt acts. hen we deal wth the eternal acton of a force on a rgd bod, the force ma be appled at an pont along ts lne of acton wthout changng ts effect on the bod as a whole and hence, consdered as a sldng vector.

6 A fed vector s one for whch a unque pont of applcaton s specfed and therefore the vector occupes a partcular poston n space. The acton of a force on a deformable bod must be specfed b a fed vector.

7 Prncple of Transmssblt The eternal effect of a force on a rgd bod wll reman unchanged f the force s moved to act on ts lne of acton. In other words, a force ma be appled at an pont on ts gven lne of acton wthout alterng the resultant eternal effects on the rgd bod on whch t acts.

8 Equalt and Equvalence of ectors Two vectors are equal f the have the same dmensons, magntudes and drectons. Two vectors are equvalent n a certan capact f each produces the ver same effect n ths capact.

9 PROPERTIES OF ECTORS Addton of ectors s done accordng to the parallelogram law of vector addton. ( ) ( ) M M or

10 Subtracton of ectors s done accordng to the parallelogram law. ( ) Z Z Multplcaton of a Scalar and a ector a a a ( b ) ab ( a b) a b a( ) a a

11 nt ector A unt vector s a free vector havng a magntude of 1 (one) as n ( n or e) It descrbes drecton. The most convenent wa to descrbe a vector n a certan drecton s to multpl ts magntude wth ts unt vector. n n 1 and have the same unt, hence the unt vector s dmensonless. Therefore, ma be epressed n terms of both ts magntude and drecton separatel. (a scalar) epresses the magntude and n (a dmensonless vector) epresses the drectonal sense of.

12 ector Components and Resultant ector Let the sum of and be. Here, and are named as the components and s named as the resultant. Sne theorem sn β snα sn γ Cosne theorem cosγ

13 The relatonshp between a force and ts vector components must not be confused wth the relatonshp between a force and ts perpendcular (orthogonal) proectons onto the same aes. For eample, the perpendcular proectons of force F onto aes a and b are F a and F b, whch are parallel to the vector components of F 1 and F 2. a b F 1 a F //a //b F a F a F 2 b F b b Components: F 1 and F 2 Proectons: F a and F b

14 It s seen that the components of a vector are not necessarl equal to the proectons of the vector onto the same aes. The components and proectons of F are equal onl when the aes a and b are perpendcular. F 1 a F //a //b F a F a b a F 2 b F b b Components: F 1 and F 2 Proectons: F a and F b

15 Cartesan Coordnates Cartesan Coordnate Sstem s composed of 90 (orthogonal) aes. It conssts of and aes n two dmensonal (planar) case,, and aes n three dmensonal (spatal) case. - aes are generall taen wthn the plane of the paper, ther postve drectons can be selected arbtrarl; the postve drecton of the aes must be determned n accordance wth the rght hand rule.

16

17 ector Components n Two Dmensonal (Planar) Cartesan Coordnates unt vector along the as,, unt vector along the as, ( ) ( ) tan θ 2 2 θ

18 ector Components n Three Dmensonal (Spatal) Cartesan Coordnates ( ) ( ) ( ) unt vector along the as,, unt vector along the as,, unt vector along the as,, ( ) 2 2 2

19 Poston ector: It s the vector that descrbes the locaton of one pont wth respect to another pont. In two dmensonal case B ( B, B ) r B/A A ( A, A ) r B/A ( ) ( ) B A B A

20 In three dmensonal case B ( B, B, B ) r B/A ( ) ( ) ( ) B r B/A A ( A, A, A ) A B A B A

21 Dot (Scalar) Product A scalar quantt s obtaned from the dot product of two vectors. cos cos a s rrelevant multplcaton of order a α α,, cos,, cos α In terms of unt vectors n Cartesan Coordnates;

22 Normal and Parallel Components of a ector wth respect to a Lne λ n // θ Magntude of parallel component: // cosθ n n cosθ cosθ 1 Parallel component:, // // n ( n) n Normal (Orthogonal) component: //

23 Cross (ector) Product: The multplcaton of two vectors n cross product results n a vector. Ths multplcaton vector s normal to the plane contanng the other two vectors. Its drecton s determned b the rght hand rule. Its magntude equals the area of the parallelogram that the vectors span. The order of multplcaton s mportant. θ a, snθ ( ) ( a ) ( a ) ( Y ) Y snθ θ

24 ,,,, sn,, sn In terms of unt vectors n Cartesan Coordnates;

25 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] [ ] [ ]

26 Med Trple Product: It s used when tang the moment of a force about a lne. ( ) ( ) ( ) or

21 Vectors: The Cross Product & Torque

21 Vectors: The Cross Product & Torque 21 Vectors: The Cross Product & Torque Do not use our left hand when applng ether the rght-hand rule for the cross product of two vectors dscussed n ths chapter or the rght-hand rule for somethng curl

More information

Goals Rotational quantities as vectors. Math: Cross Product. Angular momentum

Goals Rotational quantities as vectors. Math: Cross Product. Angular momentum Physcs 106 Week 5 Torque and Angular Momentum as Vectors SJ 7thEd.: Chap 11.2 to 3 Rotatonal quanttes as vectors Cross product Torque expressed as a vector Angular momentum defned Angular momentum as a

More information

where the coordinates are related to those in the old frame as follows.

where the coordinates are related to those in the old frame as follows. Chapter 2 - Cartesan Vectors and Tensors: Ther Algebra Defnton of a vector Examples of vectors Scalar multplcaton Addton of vectors coplanar vectors Unt vectors A bass of non-coplanar vectors Scalar product

More information

Rotation Kinematics, Moment of Inertia, and Torque

Rotation Kinematics, Moment of Inertia, and Torque Rotaton Knematcs, Moment of Inerta, and Torque Mathematcally, rotaton of a rgd body about a fxed axs s analogous to a lnear moton n one dmenson. Although the physcal quanttes nvolved n rotaton are qute

More information

Chapter 11 Torque and Angular Momentum

Chapter 11 Torque and Angular Momentum Chapter 11 Torque and Angular Momentum I. Torque II. Angular momentum - Defnton III. Newton s second law n angular form IV. Angular momentum - System of partcles - Rgd body - Conservaton I. Torque - Vector

More information

Recurrence. 1 Definitions and main statements

Recurrence. 1 Definitions and main statements Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.

More information

Laws of Electromagnetism

Laws of Electromagnetism There are four laws of electromagnetsm: Laws of Electromagnetsm The law of Bot-Savart Ampere's law Force law Faraday's law magnetc feld generated by currents n wres the effect of a current on a loop of

More information

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by 6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng

More information

Homework: 49, 56, 67, 60, 64, 74 (p. 234-237)

Homework: 49, 56, 67, 60, 64, 74 (p. 234-237) Hoework: 49, 56, 67, 60, 64, 74 (p. 34-37) 49. bullet o ass 0g strkes a ballstc pendulu o ass kg. The center o ass o the pendulu rses a ertcal dstance o c. ssung that the bullet reans ebedded n the pendulu,

More information

Q3.8: A person trying to throw a ball as far as possible will run forward during the throw. Explain why this increases the distance of the throw.

Q3.8: A person trying to throw a ball as far as possible will run forward during the throw. Explain why this increases the distance of the throw. Problem Set 3 Due: 09/3/, Tuesda Chapter 3: Vectors and Moton n Two Dmensons Questons: 7, 8,, 4, 0 Eercses & Problems:, 7, 8, 33, 37, 44, 46, 65, 73 Q3.7: An athlete performn the lon jump tres to acheve

More information

v a 1 b 1 i, a 2 b 2 i,..., a n b n i.

v a 1 b 1 i, a 2 b 2 i,..., a n b n i. SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are

More information

Support Vector Machines

Support Vector Machines Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.

More information

Review C: Work and Kinetic Energy

Review C: Work and Kinetic Energy MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department o Physcs 8.2 Revew C: Work and Knetc Energy C. Energy... 2 C.. The Concept o Energy... 2 C..2 Knetc Energy... 3 C.2 Work and Power... 4 C.2. Work Done by

More information

Faraday's Law of Induction

Faraday's Law of Induction Introducton Faraday's Law o Inducton In ths lab, you wll study Faraday's Law o nducton usng a wand wth col whch swngs through a magnetc eld. You wll also examne converson o mechanc energy nto electrc energy

More information

Politecnico di Torino. Porto Institutional Repository

Politecnico di Torino. Porto Institutional Repository Poltecnco d orno Porto Insttutonal Repostory [Proceedng] rbt dynamcs and knematcs wth full quaternons rgnal Ctaton: Andres D; Canuto E. (5). rbt dynamcs and knematcs wth full quaternons. In: 16th IFAC

More information

Description of the Force Method Procedure. Indeterminate Analysis Force Method 1. Force Method con t. Force Method con t

Description 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

What is Candidate Sampling

What is Candidate Sampling What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble

More information

CHAPTER 8 Potential Energy and Conservation of Energy

CHAPTER 8 Potential Energy and Conservation of Energy CHAPTER 8 Potental Energy and Conservaton o Energy One orm o energy can be converted nto another orm o energy. Conservatve and non-conservatve orces Physcs 1 Knetc energy: Potental energy: Energy assocated

More information

Introduction to Statistical Physics (2SP)

Introduction to Statistical Physics (2SP) Introducton to Statstcal Physcs (2SP) Rchard Sear March 5, 20 Contents What s the entropy (aka the uncertanty)? 2. One macroscopc state s the result of many many mcroscopc states.......... 2.2 States wth

More information

The OC Curve of Attribute Acceptance Plans

The OC Curve of Attribute Acceptance Plans The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4

More information

Lagrangian Dynamics: Virtual Work and Generalized Forces

Lagrangian Dynamics: Virtual Work and Generalized Forces Admssble Varatons/Vrtual Dsplacements 1 2.003J/1.053J Dynamcs and Control I, Sprng 2007 Paula Echeverr, Professor Thomas Peacock 4/4/2007 Lecture 14 Lagrangan Dynamcs: Vrtual Work and Generalzed Forces

More information

IT09 - Identity Management Policy

IT09 - Identity Management Policy IT09 - Identty Management Polcy Introducton 1 The Unersty needs to manage dentty accounts for all users of the Unersty s electronc systems and ensure that users hae an approprate leel of access to these

More information

NON-CONSTANT SUM RED-AND-BLACK GAMES WITH BET-DEPENDENT WIN PROBABILITY FUNCTION LAURA PONTIGGIA, University of the Sciences in Philadelphia

NON-CONSTANT SUM RED-AND-BLACK GAMES WITH BET-DEPENDENT WIN PROBABILITY FUNCTION LAURA PONTIGGIA, University of the Sciences in Philadelphia To appear n Journal o Appled Probablty June 2007 O-COSTAT SUM RED-AD-BLACK GAMES WITH BET-DEPEDET WI PROBABILITY FUCTIO LAURA POTIGGIA, Unversty o the Scences n Phladelpha Abstract In ths paper we nvestgate

More information

Chapter 9. Linear Momentum and Collisions

Chapter 9. Linear Momentum and Collisions Chapter 9 Lnear Momentum and Collsons CHAPTER OUTLINE 9.1 Lnear Momentum and Its Conservaton 9.2 Impulse and Momentum 9.3 Collsons n One Dmenson 9.4 Two-Dmensonal Collsons 9.5 The Center of Mass 9.6 Moton

More information

HALL EFFECT SENSORS AND COMMUTATION

HALL EFFECT SENSORS AND COMMUTATION OEM770 5 Hall Effect ensors H P T E R 5 Hall Effect ensors The OEM770 works wth three-phase brushless motors equpped wth Hall effect sensors or equvalent feedback sgnals. In ths chapter we wll explan how

More information

GRAVITY DATA VALIDATION AND OUTLIER DETECTION USING L 1 -NORM

GRAVITY DATA VALIDATION AND OUTLIER DETECTION USING L 1 -NORM GRAVITY DATA VALIDATION AND OUTLIER DETECTION USING L 1 -NORM BARRIOT Jean-Perre, SARRAILH Mchel BGI/CNES 18.av.E.Beln 31401 TOULOUSE Cedex 4 (France) Emal: jean-perre.barrot@cnes.fr 1/Introducton The

More information

Extending Probabilistic Dynamic Epistemic Logic

Extending Probabilistic Dynamic Epistemic Logic Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σ-algebra: a set

More information

BERNSTEIN POLYNOMIALS

BERNSTEIN POLYNOMIALS On-Lne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful

More information

Physics 110 Spring 2006 2-D Motion Problems: Projectile Motion Their Solutions

Physics 110 Spring 2006 2-D Motion Problems: Projectile Motion Their Solutions Physcs 110 Sprn 006 -D Moton Problems: Projectle Moton Ther Solutons 1. A place-kcker must kck a football from a pont 36 m (about 40 yards) from the oal, and half the crowd hopes the ball wll clear the

More information

Damage detection in composite laminates using coin-tap method

Damage detection in composite laminates using coin-tap method Damage detecton n composte lamnates usng con-tap method S.J. Km Korea Aerospace Research Insttute, 45 Eoeun-Dong, Youseong-Gu, 35-333 Daejeon, Republc of Korea yaeln@kar.re.kr 45 The con-tap test has the

More information

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy 4.02 Quz Solutons Fall 2004 Multple-Choce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multple-choce questons. For each queston, only one of the answers s correct.

More information

Loop Parallelization

Loop Parallelization - - Loop Parallelzaton C-52 Complaton steps: nested loops operatng on arrays, sequentell executon of teraton space DECLARE B[..,..+] FOR I :=.. FOR J :=.. I B[I,J] := B[I-,J]+B[I-,J-] ED FOR ED FOR analyze

More information

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..

More information

Face Verification Problem. Face Recognition Problem. Application: Access Control. Biometric Authentication. Face Verification (1:1 matching)

Face Verification Problem. Face Recognition Problem. Application: Access Control. Biometric Authentication. Face Verification (1:1 matching) Face Recognton Problem Face Verfcaton Problem Face Verfcaton (1:1 matchng) Querymage face query Face Recognton (1:N matchng) database Applcaton: Access Control www.vsage.com www.vsoncs.com Bometrc Authentcaton

More information

Documentation about calculation methods used for the electricity supply price index (SPIN 35.1),

Documentation about calculation methods used for the electricity supply price index (SPIN 35.1), STATISTICS SWEDEN Documentaton (6) ES/PR-S 0-- artn Kullendorff arcus rdén Documentaton about calculaton methods used for the electrct suppl prce ndex (SPIN 35.), home sales (HPI) The ndex fgure for electrct

More information

+ + + - - This circuit than can be reduced to a planar circuit

+ + + - - This circuit than can be reduced to a planar circuit MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to

More information

Distributed Multi-Target Tracking In A Self-Configuring Camera Network

Distributed Multi-Target Tracking In A Self-Configuring Camera Network Dstrbuted Mult-Target Trackng In A Self-Confgurng Camera Network Crstan Soto, B Song, Amt K. Roy-Chowdhury Department of Electrcal Engneerng Unversty of Calforna, Rversde {cwlder,bsong,amtrc}@ee.ucr.edu

More information

Rotation and Conservation of Angular Momentum

Rotation and Conservation of Angular Momentum Chapter 4. Rotaton and Conservaton of Angular Momentum Notes: Most of the materal n ths chapter s taken from Young and Freedman, Chaps. 9 and 0. 4. Angular Velocty and Acceleraton We have already brefly

More information

Reporting Forms ARF 113.0A, ARF 113.0B, ARF 113.0C and ARF 113.0D FIRB Corporate (including SME Corporate), Sovereign and Bank Instruction Guide

Reporting Forms ARF 113.0A, ARF 113.0B, ARF 113.0C and ARF 113.0D FIRB Corporate (including SME Corporate), Sovereign and Bank Instruction Guide Reportng Forms ARF 113.0A, ARF 113.0B, ARF 113.0C and ARF 113.0D FIRB Corporate (ncludng SME Corporate), Soveregn and Bank Instructon Gude Ths nstructon gude s desgned to assst n the completon of the FIRB

More information

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ). REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or

More information

Addition and Subtraction of Vectors

Addition and Subtraction of Vectors ddition and Subtraction of Vectors 1 ppendi ddition and Subtraction of Vectors In this appendi the basic elements of vector algebra are eplored. Vectors are treated as geometric entities represented b

More information

SprayExpo 2.0. Program description. SprayExpo has been developed by

SprayExpo 2.0. Program description. SprayExpo has been developed by SpraExpo.0 Program descrpton SpraExpo has been developed b Fraunhofer Insttut für Toxkologe und Expermentelle Medzn Nkola-Fuchs-Straße 1 D-3065 Hannover on behalf of the Bundesanstalt für Arbetsschutz

More information

COMPONENTS OF VECTORS

COMPONENTS OF VECTORS COMPONENTS OF VECTORS To describe motion in two dimensions we need a coordinate sstem with two perpendicular aes, and. In such a coordinate sstem, an vector A can be uniquel decomposed into a sum of two

More information

5.74 Introductory Quantum Mechanics II

5.74 Introductory Quantum Mechanics II MIT OpenCourseWare http://ocw.mt.edu 5.74 Introductory Quantum Mechancs II Sprng 9 For nformaton about ctng these materals or our Terms of Use, vst: http://ocw.mt.edu/terms. 4-1 4.1. INTERACTION OF LIGHT

More information

We are now ready to answer the question: What are the possible cardinalities for finite fields?

We are now ready to answer the question: What are the possible cardinalities for finite fields? Chapter 3 Fnte felds We have seen, n the prevous chapters, some examples of fnte felds. For example, the resdue class rng Z/pZ (when p s a prme) forms a feld wth p elements whch may be dentfed wth the

More information

Multi-Robot Tracking of a Moving Object Using Directional Sensors

Multi-Robot Tracking of a Moving Object Using Directional Sensors Mult-Robot Trackng of a Movng Object Usng Drectonal Sensors Xaomng Hu, Karl H. Johansson, Manuel Mazo Jr., Alberto Speranzon Dept. of Sgnals, Sensors & Systems Royal Insttute of Technology, SE- 44 Stockholm,

More information

1 What is a conservation law?

1 What is a conservation law? MATHEMATICS 7302 (Analytcal Dynamcs) YEAR 2015 2016, TERM 2 HANDOUT #6: MOMENTUM, ANGULAR MOMENTUM, AND ENERGY; CONSERVATION LAWS In ths handout we wll develop the concepts of momentum, angular momentum,

More information

Section V.2: Magnitudes, Directions, and Components of Vectors

Section V.2: Magnitudes, Directions, and Components of Vectors Section V.: Magnitudes, Directions, and Components of Vectors Vectors in the plane If we graph a vector in the coordinate plane instead of just a grid, there are a few things to note. Firstl, directions

More information

A Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy S-curve Regression

A Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy S-curve Regression Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy S-curve Regresson Cheng-Wu Chen, Morrs H. L. Wang and Tng-Ya Hseh Department of Cvl Engneerng, Natonal Central Unversty,

More information

Power-of-Two Policies for Single- Warehouse Multi-Retailer Inventory Systems with Order Frequency Discounts

Power-of-Two Policies for Single- Warehouse Multi-Retailer Inventory Systems with Order Frequency Discounts Power-of-wo Polces for Sngle- Warehouse Mult-Retaler Inventory Systems wth Order Frequency Dscounts José A. Ventura Pennsylvana State Unversty (USA) Yale. Herer echnon Israel Insttute of echnology (Israel)

More information

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt.

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt. Chapter 9 Revew problems 9.1 Interest rate measurement Example 9.1. Fund A accumulates at a smple nterest rate of 10%. Fund B accumulates at a smple dscount rate of 5%. Fnd the pont n tme at whch the forces

More information

Dot product and vector projections (Sect. 12.3) There are two main ways to introduce the dot product

Dot product and vector projections (Sect. 12.3) There are two main ways to introduce the dot product Dot product and vector projections (Sect. 12.3) Two definitions for the dot product. Geometric definition of dot product. Orthogonal vectors. Dot product and orthogonal projections. Properties of the dot

More information

Project Networks With Mixed-Time Constraints

Project Networks With Mixed-Time Constraints Project Networs Wth Mxed-Tme Constrants L Caccetta and B Wattananon Western Australan Centre of Excellence n Industral Optmsaton (WACEIO) Curtn Unversty of Technology GPO Box U1987 Perth Western Australa

More information

The Use of Analytics for Claim Fraud Detection Roosevelt C. Mosley, Jr., FCAS, MAAA Nick Kucera Pinnacle Actuarial Resources Inc.

The Use of Analytics for Claim Fraud Detection Roosevelt C. Mosley, Jr., FCAS, MAAA Nick Kucera Pinnacle Actuarial Resources Inc. Paper 1837-2014 The Use of Analytcs for Clam Fraud Detecton Roosevelt C. Mosley, Jr., FCAS, MAAA Nck Kucera Pnnacle Actuaral Resources Inc., Bloomngton, IL ABSTRACT As t has been wdely reported n the nsurance

More information

Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits

Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.

More information

NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582

NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582 NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582 7. Root Dynamcs 7.2 Intro to Root Dynamcs We now look at the forces requred to cause moton of the root.e. dynamcs!!

More information

Nordea G10 Alpha Carry Index

Nordea G10 Alpha Carry Index Nordea G10 Alpha Carry Index Index Rules v1.1 Verson as of 10/10/2013 1 (6) Page 1 Index Descrpton The G10 Alpha Carry Index, the Index, follows the development of a rule based strategy whch nvests and

More information

Elastic Systems for Static Balancing of Robot Arms

Elastic Systems for Static Balancing of Robot Arms . th World ongress n Mechans and Machne Scence, Guanajuato, Méco, 9- June, 0 _ lastc Sstes for Statc alancng of Robot rs I.Sonescu L. uptu Lucana Ionta I.Ion M. ne Poltehnca Unverst Poltehnca Unverst Poltehnca

More information

"Research Note" APPLICATION OF CHARGE SIMULATION METHOD TO ELECTRIC FIELD CALCULATION IN THE POWER CABLES *

Research Note APPLICATION OF CHARGE SIMULATION METHOD TO ELECTRIC FIELD CALCULATION IN THE POWER CABLES * Iranan Journal of Scence & Technology, Transacton B, Engneerng, ol. 30, No. B6, 789-794 rnted n The Islamc Republc of Iran, 006 Shraz Unversty "Research Note" ALICATION OF CHARGE SIMULATION METHOD TO ELECTRIC

More information

The Mathematical Derivation of Least Squares

The Mathematical Derivation of Least Squares Pscholog 885 Prof. Federco The Mathematcal Dervaton of Least Squares Back when the powers that e forced ou to learn matr algera and calculus, I et ou all asked ourself the age-old queston: When the hell

More information

sin(θ) = opp hyp cos(θ) = adj hyp tan(θ) = opp adj

sin(θ) = opp hyp cos(θ) = adj hyp tan(θ) = opp adj Math, Trigonometr and Vectors Geometr 33º What is the angle equal to? a) α = 7 b) α = 57 c) α = 33 d) α = 90 e) α cannot be determined α Trig Definitions Here's a familiar image. To make predictive models

More information

Calculation of Sampling Weights

Calculation of Sampling Weights Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a two-stage stratfed cluster desgn. 1 The frst stage conssted of a sample

More information

A machine vision approach for detecting and inspecting circular parts

A machine vision approach for detecting and inspecting circular parts A machne vson approach for detectng and nspectng crcular parts Du-Mng Tsa Machne Vson Lab. Department of Industral Engneerng and Management Yuan-Ze Unversty, Chung-L, Tawan, R.O.C. E-mal: edmtsa@saturn.yzu.edu.tw

More information

Math, Trigonometry and Vectors. Geometry. Trig Definitions. sin(θ) = opp hyp. cos(θ) = adj hyp. tan(θ) = opp adj. Here's a familiar image.

Math, Trigonometry and Vectors. Geometry. Trig Definitions. sin(θ) = opp hyp. cos(θ) = adj hyp. tan(θ) = opp adj. Here's a familiar image. Math, Trigonometr and Vectors Geometr Trig Definitions Here's a familiar image. To make predictive models of the phsical world, we'll need to make visualizations, which we can then turn into analtical

More information

O(n) mass matrix inversion for serial manipulators and polypeptide chains using Lie derivatives Kiju Lee, Yunfeng Wang and Gregory S.

O(n) mass matrix inversion for serial manipulators and polypeptide chains using Lie derivatives Kiju Lee, Yunfeng Wang and Gregory S. Robotca 7) volume 5, pp 739 75 7 Cambrdge Unversty Press do:7/s6357477385 Prnted n the Unted Kngdom On) mass matrx nverson for seral manpulators and polypeptde chans usng Le dervatves Ku Lee, Yunfeng Wang

More information

1 Example 1: Axis-aligned rectangles

1 Example 1: Axis-aligned rectangles COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton

More information

RESEARCH ON DUAL-SHAKER SINE VIBRATION CONTROL. Yaoqi FENG 1, Hanping QIU 1. China Academy of Space Technology (CAST) yaoqi.feng@yahoo.

RESEARCH ON DUAL-SHAKER SINE VIBRATION CONTROL. Yaoqi FENG 1, Hanping QIU 1. China Academy of Space Technology (CAST) yaoqi.feng@yahoo. ICSV4 Carns Australa 9- July, 007 RESEARCH ON DUAL-SHAKER SINE VIBRATION CONTROL Yaoq FENG, Hanpng QIU Dynamc Test Laboratory, BISEE Chna Academy of Space Technology (CAST) yaoq.feng@yahoo.com Abstract

More information

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of

More information

A Falling Detection System with wireless sensor for the Elderly People Based on Ergnomics

A Falling Detection System with wireless sensor for the Elderly People Based on Ergnomics Vol.8, o.1 (14), pp.187-196 http://dx.do.org/1.1457/sh.14.8.1. A Fallng Detecton System wth wreless sensor for the Elderly People Based on Ergnomcs Zhenhe e 1, ng L, Qaoxang Zhao and Xue Lu 3 1 College

More information

Computational Fluid Dynamics II

Computational Fluid Dynamics II Computatonal Flud Dynamcs II Eercse 2 1. Gven s the PDE: u tt a 2 ou Formulate the CFL-condton for two possble eplct schemes. 2. The Euler equatons for 1-dmensonal, unsteady flows s dscretsed n the followng

More information

DECOMPOSITION OF MEASURED GROUND VIBRATIONS INTO BASIC SOIL WAVES

DECOMPOSITION OF MEASURED GROUND VIBRATIONS INTO BASIC SOIL WAVES DECOMPOSITION OF MEASURED GROUND VIBRATIONS INTO BASIC SOIL WAVES D. Macjausas Department of Scence & Technology, Unversty of Luembourg, Luembourg S. Van Baars Department of Scence & Technology, Unversty

More information

Conversion between the vector and raster data structures using Fuzzy Geographical Entities

Conversion between the vector and raster data structures using Fuzzy Geographical Entities Converson between the vector and raster data structures usng Fuzzy Geographcal Enttes Cdála Fonte Department of Mathematcs Faculty of Scences and Technology Unversty of Combra, Apartado 38, 3 454 Combra,

More information

Recall that two vectors in are perpendicular or orthogonal provided that their dot

Recall that two vectors in are perpendicular or orthogonal provided that their dot Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal

More information

Section 5.4 Annuities, Present Value, and Amortization

Section 5.4 Annuities, Present Value, and Amortization Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today

More information

An Enhanced Super-Resolution System with Improved Image Registration, Automatic Image Selection, and Image Enhancement

An Enhanced Super-Resolution System with Improved Image Registration, Automatic Image Selection, and Image Enhancement An Enhanced Super-Resoluton System wth Improved Image Regstraton, Automatc Image Selecton, and Image Enhancement Yu-Chuan Kuo ( ), Chen-Yu Chen ( ), and Chou-Shann Fuh ( ) Department of Computer Scence

More information

Brigid Mullany, Ph.D University of North Carolina, Charlotte

Brigid Mullany, Ph.D University of North Carolina, Charlotte Evaluaton And Comparson Of The Dfferent Standards Used To Defne The Postonal Accuracy And Repeatablty Of Numercally Controlled Machnng Center Axes Brgd Mullany, Ph.D Unversty of North Carolna, Charlotte

More information

CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES

CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES In ths chapter, we wll learn how to descrbe the relatonshp between two quanttatve varables. Remember (from Chapter 2) that the terms quanttatve varable

More information

Feature selection for intrusion detection. Slobodan Petrović NISlab, Gjøvik University College

Feature selection for intrusion detection. Slobodan Petrović NISlab, Gjøvik University College Feature selecton for ntruson detecton Slobodan Petrovć NISlab, Gjøvk Unversty College Contents The feature selecton problem Intruson detecton Traffc features relevant for IDS The CFS measure The mrmr measure

More information

Vision Mouse. Saurabh Sarkar a* University of Cincinnati, Cincinnati, USA ABSTRACT 1. INTRODUCTION

Vision Mouse. Saurabh Sarkar a* University of Cincinnati, Cincinnati, USA ABSTRACT 1. INTRODUCTION Vson Mouse Saurabh Sarkar a* a Unversty of Cncnnat, Cncnnat, USA ABSTRACT The report dscusses a vson based approach towards trackng of eyes and fngers. The report descrbes the process of locatng the possble

More information

Efficient Project Portfolio as a tool for Enterprise Risk Management

Efficient Project Portfolio as a tool for Enterprise Risk Management Effcent Proect Portfolo as a tool for Enterprse Rsk Management Valentn O. Nkonov Ural State Techncal Unversty Growth Traectory Consultng Company January 5, 27 Effcent Proect Portfolo as a tool for Enterprse

More information

HÜCKEL MOLECULAR ORBITAL THEORY

HÜCKEL MOLECULAR ORBITAL THEORY 1 HÜCKEL MOLECULAR ORBITAL THEORY In general, the vast maorty polyatomc molecules can be thought of as consstng of a collecton of two electron bonds between pars of atoms. So the qualtatve pcture of σ

More information

Dr. Fritz Wilhelm, DVC,8/30/2004;4:25 PM E:\Excel files\ch 03 Vector calculations.doc Last printed 8/30/2004 4:25:00 PM

Dr. Fritz Wilhelm, DVC,8/30/2004;4:25 PM E:\Excel files\ch 03 Vector calculations.doc Last printed 8/30/2004 4:25:00 PM E:\Ecel files\ch 03 Vector calculations.doc Last printed 8/30/2004 4:25:00 PM Vector calculations 1 of 6 Vectors are ordered sequences of numbers. In three dimensions we write vectors in an of the following

More information

Consider a 1-D stationary state diffusion-type equation, which we will call the generalized diffusion equation from now on:

Consider a 1-D stationary state diffusion-type equation, which we will call the generalized diffusion equation from now on: Chapter 1 Boundary value problems Numercal lnear algebra technques can be used for many physcal problems. In ths chapter we wll gve some examples of how these technques can be used to solve certan boundary

More information

FREQUENCY OF OCCURRENCE OF CERTAIN CHEMICAL CLASSES OF GSR FROM VARIOUS AMMUNITION TYPES

FREQUENCY OF OCCURRENCE OF CERTAIN CHEMICAL CLASSES OF GSR FROM VARIOUS AMMUNITION TYPES FREQUENCY OF OCCURRENCE OF CERTAIN CHEMICAL CLASSES OF GSR FROM VARIOUS AMMUNITION TYPES Zuzanna BRO EK-MUCHA, Grzegorz ZADORA, 2 Insttute of Forensc Research, Cracow, Poland 2 Faculty of Chemstry, Jagellonan

More information

n + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2)

n + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2) MATH 16T Exam 1 : Part I (In-Class) Solutons 1. (0 pts) A pggy bank contans 4 cons, all of whch are nckels (5 ), dmes (10 ) or quarters (5 ). The pggy bank also contans a con of each denomnaton. The total

More information

Least Squares Fitting of Data

Least Squares Fitting of Data Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 1998-2016. All Rghts Reserved. Created: July 15, 1999 Last Modfed: January 5, 2015 Contents 1 Lnear Fttng

More information

1. Measuring association using correlation and regression

1. Measuring association using correlation and regression How to measure assocaton I: Correlaton. 1. Measurng assocaton usng correlaton and regresson We often would lke to know how one varable, such as a mother's weght, s related to another varable, such as a

More information

PERRON FROBENIUS THEOREM

PERRON FROBENIUS THEOREM PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, ()

More information

Section 11.4: Equations of Lines and Planes

Section 11.4: Equations of Lines and Planes Section 11.4: Equations of Lines and Planes Definition: The line containing the point ( 0, 0, 0 ) and parallel to the vector v = A, B, C has parametric equations = 0 + At, = 0 + Bt, = 0 + Ct, where t R

More information

Overview of monitoring and evaluation

Overview of monitoring and evaluation 540 Toolkt to Combat Traffckng n Persons Tool 10.1 Overvew of montorng and evaluaton Overvew Ths tool brefly descrbes both montorng and evaluaton, and the dstncton between the two. What s montorng? Montorng

More information

Lecture 2 The First Law of Thermodynamics (Ch.1)

Lecture 2 The First Law of Thermodynamics (Ch.1) Lecture he Frst Law o hermodynamcs (Ch.) Outlne:. Internal Energy, Work, Heatng. Energy Conservaton the Frst Law 3. Quas-statc processes 4. Enthalpy 5. Heat Capacty Internal Energy he nternal energy o

More information

Vibration Analysis using Time Domain Methods for the Detection of small Roller Bearing Defects

Vibration Analysis using Time Domain Methods for the Detection of small Roller Bearing Defects SIRM 9-8th Internatonal Conference on Vbratons n Rotatng Machnes, Venna, Austra, 3-5 February 9 Vbraton Analyss usng Tme Doman Methods for the Detecton of small Roller Bearng Defects Tahsn Doguer Insttut

More information

Implementation of Deutsch's Algorithm Using Mathcad

Implementation of Deutsch's Algorithm Using Mathcad Implementaton of Deutsch's Algorthm Usng Mathcad Frank Roux The followng s a Mathcad mplementaton of Davd Deutsch's quantum computer prototype as presented on pages - n "Machnes, Logc and Quantum Physcs"

More information

Realistic Image Synthesis

Realistic Image Synthesis Realstc Image Synthess - Combned Samplng and Path Tracng - Phlpp Slusallek Karol Myszkowsk Vncent Pegoraro Overvew: Today Combned Samplng (Multple Importance Samplng) Renderng and Measurng Equaton Random

More information

Lecture 2: Single Layer Perceptrons Kevin Swingler

Lecture 2: Single Layer Perceptrons Kevin Swingler Lecture 2: Sngle Layer Perceptrons Kevn Sngler kms@cs.str.ac.uk Recap: McCulloch-Ptts Neuron Ths vastly smplfed model of real neurons s also knon as a Threshold Logc Unt: W 2 A Y 3 n W n. A set of synapses

More information

w = COI EYE view direction vector u = w ( 010,, ) cross product with y-axis v = w u up vector

w = COI EYE view direction vector u = w ( 010,, ) cross product with y-axis v = w u up vector . w COI EYE view direction vector u w ( 00,, ) cross product with -ais v w u up vector (EQ ) Computer Animation: Algorithms and Techniques 29 up vector view vector observer center of interest 30 Computer

More information

Evaluation of Coordination Strategies for Heterogeneous Sensor Networks Aiming at Surveillance Applications

Evaluation of Coordination Strategies for Heterogeneous Sensor Networks Aiming at Surveillance Applications Evaluaton of Coordnaton Strateges for Heterogeneous Sensor Networs Amng at Survellance Applcatons Edson Pgnaton de Fretas, *, Tales Hemfarth*, Carlos Eduardo Perera*, Armando Morado Ferrera, Flávo Rech

More information

Figure 1.1 Vector A and Vector F

Figure 1.1 Vector A and Vector F CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have

More information

Traffic State Estimation in the Traffic Management Center of Berlin

Traffic State Estimation in the Traffic Management Center of Berlin Traffc State Estmaton n the Traffc Management Center of Berln Authors: Peter Vortsch, PTV AG, Stumpfstrasse, D-763 Karlsruhe, Germany phone ++49/72/965/35, emal peter.vortsch@ptv.de Peter Möhl, PTV AG,

More information

Safety and Reliability of Distributed Embedded Systems

Safety and Reliability of Distributed Embedded Systems Saety and Relablty o Dstrbuted Embedded Systems Techncal Report ESL 04-01 Smulaton o Vehcle Longtudnal Dynamcs Mchael Short Mchael J. Pont and Qang Huang Embedded Systems Laboratory Unversty o Lecester

More information