This circuit than can be reduced to a planar circuit

Size: px
Start display at page:

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

Transcription

1 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 wrtng a bunch of KVLs. Note: Meshcurrent method only works for planar crcuts: crcuts that can be drawn on a plane (lke on a paper) wthout any elements or connectng wres crossng each other as shown below. Note that n some cases a crcut that looks nonplanar can be made nto a planar crcut by mong some of the connectng wres (see fgure) Wres crossng Ths crcut than can be reduced to a planar crcut A Nonplanar Crcut Meshcurrent method s best explaned n the context of example crcut below. A mesh s defned as a closed path (a loop) that contans no closed path wthn t. Mesh current s the current that crculates n the mesh.e., a) f an element s located on a sngle mesh (such as R 1, R 2, s1,and s2 ) t carres the same current as the mesh current, b) If an element s located on the boundary of two meshes (such as R 3 ), t wll carry a current that s the algebrac sum of the the two mesh currents: 1 1 s1 R 1 R R s2 3 = = 2 1 In ths way, KCLs are automatcally satsfed. In addton, as we can wrte current n each element n terms of mesh currents, we can use characterstcs of element to wrte the MAE140 Notes, Wnter

2 oltage across each element n terms on mesh currents. Therefore, we need only to wrte KVLs n terms of mesh currents. In the crcut aboe, KVLs ge: Mesh 1: R 1 1 R 3 ( 1 2 ) s1 =0 (R 1 R 3 ) 1 R 3 2 = s1 Mesh 2: R 3 ( 2 1 )R 2 2 s1 =0 R 3 1 (R 2 R 3 ) 2 = s2 or n matrx form, [ R1 R 3 R 3 R 3 R 2 R 3 ] [ 1 2 ] [ s1 = s2 ] R = s Whch s smlar n form to matrx equaton found for nodeoltage method. s the array of mesh currents (unknowns), s s the array of ndependent oltage sources, and R s the resstance matrx and s symmetrc. The dagonal element, R jj, s the sum of resstance around mesh j and the offdagonal elements, R jk, are the sum of resstance shared by meshes j and k. Example Fnd and. Usng meshcurrent method: 1 2Ω 6 Ω 2 Mesh 1: ( 1 2 ) 16 = 0 Mesh 2: ( 2 1 ) 9=0 { = =3 16 V 9 V 1 2 3Ω 6 V { 1 =2A 2 =1A The problem unknowns, and can now be found from the mesh currents: = 1 2 =1A =2 1 =4V Mesh currents method for crcuts wth current sources Because of characterstcs of a current source does not specfy ts oltage, we hae to modfy meshcurrent method. Ths s best seen n the example below: MAE140 Notes, Wnter

3 From the crcut, we note: If a current source s located on only one mesh (1A ICS n the crcut), the mesh current can be drectly found from the current source and we do not need to wrte any KVL: 1 = 1 A 1 A 1 2Ω 2 10 V 2A 2Ω 3 6Ω If a current source s located on the boundary between two meshes (2A ICS n the crcut), KVL on these meshes (mesh 2 or 3 n the aboe crcut) contan the oltage across the 2A ICS whch s unknown. We need two equatons to substtue for the two KVLs on meshes 2 and 3 that are not useful now. The frst one s found from the characterstcs of the current source (ts current should be 2 A): 3 2 =2A The second equaton can be found by notng that KVL can be wrtten oer any closed loop. We defne a supermesh as the combnaton of two meshes whch hae a current source on ther boundary as shown n the fgure. Whle KVL on mesh 2 or on mesh 3 both nclude the oltage across the 2A current source that s unknown, KVL on the supermesh does not nclude that: Supermesh 2&3: 2( 2 1 )2( 3 1 ) = =10 1 =1 2 3 = = =6 1 =1 2 = 3 2 2( 3 2) 8 3 = 10 whch results n 1 =1A, 2 = 1 A,and 3 =1A, 1 A Note that we could hae used 1 =1and 2 = 3 2 equatons drectly on the meshes as shown n the fgure and wrote only the KVL on the supermesh: 2Ω 1 = 1 2Ω 2 = V 2A 3 6Ω supermesh 2&3: 2[( 3 2) (1)) 2[ 3 (1)] = 0 3 =1A 2 =1 2=1 A MAE140 Notes, Wnter

4 Recpe for MeshCurrent Method 1. Checkf crcut s planar. 2. Identfy meshes, mesh currents, and supermeshes. a) Rearrange the crcut f possble to poston current source on a sngle mesh. b) Use characterstc equatons of ICS to fnd mesh currents and reduce the number of unknowns. 3. Wrte KVL at each mesh and supermesh. 4. Sole for mesh currents. 5. Calculate problem unknowns from mesh currents. If you need to calculate the oltage across a current source you may hae to wrte KVL around a mesh contanng the current source. 6. For consstency and elmnaton of errors, always markall mesh currents n clockwse drecton and wrte down KVLs n the same drecton. Comparson of Nodeoltage and Meshcurrent methods Nodeoltage and meshcurrent are powerful methods that smplfy crcut analyss substantally. They are methods of choce n almost all cases (except for ery smple crcuts or specal crcuts). Examnaton of the crcut can also tell us whch of the two methods are best suted for the crcut at hand. We always want to reduce the crcut equatons nto the smallest number of equatons n smallest number of unknowns. The number of equatons from nodeoltage method, N NV and mesh current method, N MC are gen by: N NV = N node 1 N VS N MC = N mesh N CS where N VS and N CS are numbers of oltage and current sources, respectely. Thus, always nspect the crcut, fnd N VS and N CS, and proceed wth the method that results n the smallest number of equatons to sole. Note: You need to checkto ensure that the crcut s a planar crcut. If t s not one cannot use meshcurrent method and should use nodeoltage method. MAE140 Notes, Wnter

5 Addte propertes and Superposton In solng any lnear crcut, we always end up wth a set of smultaneous lnear equatons of the form A x = s A : x : s : matrx of resstances or conductances array of crcut arables, and/or, (unknowns) array of ndependent sources Lnear algebra tells that f we know the soluton to A x = s 1 to be x 1 (.e., A x 1 = s 1 ) and f we know the soluton to A x = s 2 to be x 2 (.e., A x 2 = s 2 ), then the soluton to A x = s 1 s 2 s x = x 1 x 2 because: A x 1 A x 2 = s 1 s 2 A (x 1 x 2 )=s 1 s 2 In a lnear crcut ths property means that: Addte property of lnear crcut or Prncple of Superposton: If a lnear crcut s dren by more than one ndependent source, the response of the crcut can be wrtten as the sum of the responses of the crcut to nddual sources wth all other sources klled (.e., ther strength set to zero.) Note that kllng a source does not mean remoeng t. s s = s for any klled = 0 for any = s klled = 0 for any for any MAE140 Notes, Wnter

6 Example: Fnd by superposton. Because we hae two ndependent sources, we frst kll the current source to arre at crcut a and then we kll the oltage source to arre at crcut b. By superposton, = a b Crcut a s a oltage dder crcut and a can be wrtten down drectly as 10 Ω 5Ω 3 A 15 V 10 Ω a = = 5 V Crcut b s a current dder crcut and current can be wrtten down drectly as a 5Ω 10 Ω 15 V = 1/51/10 3=2A 1/5 b = 5 = 10 V b 5Ω 3 A Thus, = a b =5 10 = 5 V. Note: Usng superposton results n slghtly smpler crcuts (one element s replaced wth ether a short or open crcut) but more crcuts. In general superposton requres more work than nodeoltage or meshcurrent methods. Superposton s used: a) If sources are fundamentally dfferent (e.g., dc and ac sources as we see later). In ths case superposton may be the only choce, b) If crcut s repette (see example 312 text book) such that crcuts resultng from applyng superposton lookdentcal and, thus, we need only to sole one crcut. MAE140 Notes, Wnter

7 Reducton of twotermnal subcrcuts to Theenn form Recall Theenn and Norton forms and the R T fact that they are equalent. The conenton s to wrte the characterstcs of N Theenn/Norton forms wth acte sgn conenton: T RN = T R T = N R Theenn Form Norton Form N Equalent f R =R T N and = R T N T We used the equalency of Norton and Theenn forms n crcut reducton. Recall our dscusson of equalent elements and subcrcuts. We can replace any twotermnal subcrcut wth another one as long as they hae the same characterstcs. We wll show below that the characterstcs of any twotermnal element contanng lnear elements s n Theenn form. Frst let s examne how to fnd characterstcs of a twotermnal element. In order to fnd the characterstcs of a resstor n the lab, we connect a oltage sources (wth adjustable strength) to ts termnal, change the strength of the oltage of the source and Subcrcut measure the current flowng through the resstor. After a suffcent number of pars of and are measured, we can plot the result and deduce = R. We can perform smlar, but mathematcal, experment to fnd the characterstcs of a twotermnal element. Attach a oltage source across ts termnal wth a strength. Sole the crcut and calculate current Subcrcut whch wll be n terms of ( characterstcs!). Alternately, we can attach a current source wth strength to the subcrcut and sole for n terms as s shown. Ths s the general method to calculate the characterstcs of a twotermnal element. Suppose we used nodeoltage method and assgn the ground as shown. After wrtng all of nodeoltage equatons, we wll get: G = s G : : s : Conductance matrx Array of node oltages Array of ndependent current sources If we choose as node no. 1 to be the poste termnal of the subcrcut, the nodeoltage array wll be a column ector, =[, 2, 3,..] wth oltage as ts frst element. The MAE140 Notes, Wnter

8 current source array s also wll hae as ts frst element. Now, f we sole the aboe matrx equaton and denotng the nerse matrx of G as G 1,weget: 2... n = G 1 11 G G1n 1 G 1 21 G G2n G 1 n1 G 1 n2... G 1 nn s2... sn The frst row of ths matrx equaton reduces to = C 1 C 2 where C 1 and C 2 are two constants (Snce all G and s are constants). C 1 should be a resstance (call t R T )and C 2 should be a oltage (call t V T ). Thus: the characterstcs of any twotermnal element contanng lnear elements s n the Theenn form of = T R T. Next, consder the twotermnal subcrcut = 0 R T = 0 and ts Theenn equalent (they hae exactly same characterstcs). Let the cur T Subcrcut = rent = 0 and calculate,.e., calculate the oc = oc oltage across the termnals of the subcrcut whle the termnal are open crcut. Ths oltage s called the open crcut oltage, oc.examnaton of the Theenn form shows that f =0, T = oc. Next, consder the twotermnal subcrcut and ts Norton equalent (they hae exactly same characterstcs). Let the oltage = 0 and calculate,.e., calculate the current whle the subcrcut termnals are shorted. Ths current s called the short crcut current, sc. Examnaton of the Norton form shows that f =0, N = sc. Subcrcut = sc = 0 N R N = sc Lastly, examnaton of the matrx equaton aboe shows that the Theenn resstance depends of conductance matrx only. Thus, f one klls all of the sources n the subcrcut, the remanng crcut should be equalent to the Theenn s resstance. The aboe bold/underlned statements consttute the Theenn s Theorem Fndng Equalent Theenn/Norton Forms: Three methods are aalable: Method 1: Use source transformaton and crcut reducton to reduce the crcut to a Theenn/Norton form. Ths s a cumbersome method, does not always work, and should be used only on smple crcuts. MAE140 Notes, Wnter = 0

9 Method 2: Drectly fnd characterstcs of the subcrcut by attachng a current source or a oltage source to the crcut as dscussed aboe. Ths method always work. The drawback s that the crcut has to be soled analytcally. Method 3: Compute two of the followng quanttes by solng the approprate crcuts: T = oc, N = sc, and R T by kllng the sources. The thrd parameter s found from T = R T N. Ths s the best method and wth a few exceptons, always work. Example: Fnd Theenn equalent of ths subcrcut: 5 Ω 20Ω Method 1: Source transformaton and crcut reducton 5 Ω 5A 5 Ω 20Ω 20Ω 4 Ω 8 Ω 8A 32V 32V Method 2: Drectly fnd characterstcs. We attach a oltage source wth strength to the output termnals as shown. Assume s known and sole for. Crcut has 4 nodes and two oltage sources, so the number of equaton for nodeoltage method, N NV =4 1 2=1. Crcut has three meshes and 1 current source, so the number of equatons for meshcurrent method s N MC =3 1 = 2. So, better to do nodeoltage method. 5 Ω 1 20Ω MAE140 Notes, Wnter

10 =160.5 = = 4 4 T =32V R T =8Ω = = 0 =32 8 ( characterstcs!) Method 3: Theenn s Theorem: fnd two of the followng three: R T, oc,and sc. a) Fnd R T by kllng the sources 5 Ω 5 Ω 20Ω 20Ω 5 20 = R = 44 = 8Ω T b) Fnd T = oc (set =0) Usng nodeoltage method (note that the oltage drop across the 4 Ω resstor s zero. oc 25 oc 0 3= oc 100 oc 60 = 0 5 Ω 20 Ω oc = 0 oc T = oc =32V c) Fnd N = sc (set =0) Usng nodeoltage method: =0 =0 5 Ω 20Ω 2 = 0 sc 2 =16V N = sc = 2 4 =4A MAE140 Notes, Wnter

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

(6)(2) (-6)(-4) (-4)(6) + (-2)(-3) + (4)(3) + (2)(-3) = -12-24 + 24 + 6 + 12 6 = 0

(6)(2) (-6)(-4) (-4)(6) + (-2)(-3) + (4)(3) + (2)(-3) = -12-24 + 24 + 6 + 12 6 = 0 Chapter 3 Homework Soluton P3.-, 4, 6, 0, 3, 7, P3.3-, 4, 6, P3.4-, 3, 6, 9, P3.5- P3.6-, 4, 9, 4,, 3, 40 ---------------------------------------------------- P 3.- Determne the alues of, 4,, 3, and 6

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

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

The Full-Wave Rectifier

The Full-Wave Rectifier 9/3/2005 The Full Wae ectfer.doc /0 The Full-Wae ectfer Consder the followng juncton dode crcut: s (t) Power Lne s (t) 2 Note that we are usng a transformer n ths crcut. The job of ths transformer s to

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

The circuit shown on Figure 1 is called the common emitter amplifier circuit. The important subsystems of this circuit are:

The circuit shown on Figure 1 is called the common emitter amplifier circuit. The important subsystems of this circuit are: polar Juncton Transstor rcuts Voltage and Power Amplfer rcuts ommon mtter Amplfer The crcut shown on Fgure 1 s called the common emtter amplfer crcut. The mportant subsystems of ths crcut are: 1. The basng

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

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

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

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

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

Multiple stage amplifiers

Multiple stage amplifiers Multple stage amplfers Ams: Examne a few common 2-transstor amplfers: -- Dfferental amplfers -- Cascode amplfers -- Darlngton pars -- current mrrors Introduce formal methods for exactly analysng multple

More information

Finite Math Chapter 10: Study Guide and Solution to Problems

Finite Math Chapter 10: Study Guide and Solution to Problems Fnte Math Chapter 10: Study Gude and Soluton to Problems Basc Formulas and Concepts 10.1 Interest Basc Concepts Interest A fee a bank pays you for money you depost nto a savngs account. Prncpal P The amount

More information

Section 5.3 Annuities, Future Value, and Sinking Funds

Section 5.3 Annuities, Future Value, and Sinking Funds Secton 5.3 Annutes, Future Value, and Snkng Funds Ordnary Annutes A sequence of equal payments made at equal perods of tme s called an annuty. The tme between payments s the payment perod, and the tme

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

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

The Greedy Method. Introduction. 0/1 Knapsack Problem

The Greedy Method. Introduction. 0/1 Knapsack Problem The Greedy Method Introducton We have completed data structures. We now are gong to look at algorthm desgn methods. Often we are lookng at optmzaton problems whose performance s exponental. For an optmzaton

More information

Small-Signal Analysis of BJT Differential Pairs

Small-Signal Analysis of BJT Differential Pairs 5/11/011 Dfferental Moe Sall Sgnal Analyss of BJT Dff Par 1/1 SallSgnal Analyss of BJT Dfferental Pars Now lets conser the case where each nput of the fferental par conssts of an entcal D bas ter B, an

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

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

How To Calculate The Accountng Perod Of Nequalty

How To Calculate The Accountng Perod Of Nequalty Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.

More information

Comparison of Control Strategies for Shunt Active Power Filter under Different Load Conditions

Comparison of Control Strategies for Shunt Active Power Filter under Different Load Conditions Comparson of Control Strateges for Shunt Actve Power Flter under Dfferent Load Condtons Sanjay C. Patel 1, Tushar A. Patel 2 Lecturer, Electrcal Department, Government Polytechnc, alsad, Gujarat, Inda

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

Chapter 12 Inductors and AC Circuits

Chapter 12 Inductors and AC Circuits hapter Inductors and A rcuts awrence B. ees 6. You may make a sngle copy of ths document for personal use wthout wrtten permsson. Hstory oncepts from prevous physcs and math courses that you wll need for

More information

THE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES

THE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered

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

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

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

Chapter 6 Inductance, Capacitance, and Mutual Inductance

Chapter 6 Inductance, Capacitance, and Mutual Inductance Chapter 6 Inductance Capactance and Mutual Inductance 6. The nductor 6. The capactor 6.3 Seres-parallel combnatons of nductance and capactance 6.4 Mutual nductance 6.5 Closer look at mutual nductance Oerew

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

Using Series to Analyze Financial Situations: Present Value

Using Series to Analyze Financial Situations: Present Value 2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated

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

Resistive Network Analysis. The Node Voltage Method - 1

Resistive Network Analysis. The Node Voltage Method - 1 esste Network Anlyss he nlyss of n electrcl network conssts of determnng ech of the unknown rnch currents nd node oltges. A numer of methods for network nlyss he een deeloped, sed on Ohm s Lw nd Krchoff

More information

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12 14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed

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

8 Algorithm for Binary Searching in Trees

8 Algorithm for Binary Searching in Trees 8 Algorthm for Bnary Searchng n Trees In ths secton we present our algorthm for bnary searchng n trees. A crucal observaton employed by the algorthm s that ths problem can be effcently solved when the

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

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

An Alternative Way to Measure Private Equity Performance

An Alternative Way to Measure Private Equity Performance An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate

More information

Traffic-light a stress test for life insurance provisions

Traffic-light a stress test for life insurance provisions MEMORANDUM Date 006-09-7 Authors Bengt von Bahr, Göran Ronge Traffc-lght a stress test for lfe nsurance provsons Fnansnspetonen P.O. Box 6750 SE-113 85 Stocholm [Sveavägen 167] Tel +46 8 787 80 00 Fax

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

Luby s Alg. for Maximal Independent Sets using Pairwise Independence

Luby s Alg. for Maximal Independent Sets using Pairwise Independence Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent

More information

Production. 2. Y is closed A set is closed if it contains its boundary. We need this for the solution existence in the profit maximization problem.

Production. 2. Y is closed A set is closed if it contains its boundary. We need this for the solution existence in the profit maximization problem. Producer Theory Producton ASSUMPTION 2.1 Propertes of the Producton Set The producton set Y satsfes the followng propertes 1. Y s non-empty If Y s empty, we have nothng to talk about 2. Y s closed A set

More information

Mesh-Current Method (Loop Analysis)

Mesh-Current Method (Loop Analysis) Mesh-Current Method (Loop Analysis) Nodal analysis was developed by applying KCL at each non-reference node. Mesh-Current method is developed by applying KVL around meshes in the circuit. A mesh is a loop

More information

Generalizing the degree sequence problem

Generalizing the degree sequence problem Mddlebury College March 2009 Arzona State Unversty Dscrete Mathematcs Semnar The degree sequence problem Problem: Gven an nteger sequence d = (d 1,...,d n ) determne f there exsts a graph G wth d as ts

More information

Chapter 4 ECONOMIC DISPATCH AND UNIT COMMITMENT

Chapter 4 ECONOMIC DISPATCH AND UNIT COMMITMENT Chapter 4 ECOOMIC DISATCH AD UIT COMMITMET ITRODUCTIO A power system has several power plants. Each power plant has several generatng unts. At any pont of tme, the total load n the system s met by the

More information

total A A reag total A A r eag

total A A reag total A A r eag hapter 5 Standardzng nalytcal Methods hapter Overvew 5 nalytcal Standards 5B albratng the Sgnal (S total ) 5 Determnng the Senstvty (k ) 5D Lnear Regresson and albraton urves 5E ompensatng for the Reagent

More information

7.5. Present Value of an Annuity. Investigate

7.5. Present Value of an Annuity. Investigate 7.5 Present Value of an Annuty Owen and Anna are approachng retrement and are puttng ther fnances n order. They have worked hard and nvested ther earnngs so that they now have a large amount of money on

More information

1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)

1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP) 6.3 / -- Communcaton Networks II (Görg) SS20 -- www.comnets.un-bremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes

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

How Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence

How Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence 1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh

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

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

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

VRT012 User s guide V0.1. Address: Žirmūnų g. 27, Vilnius LT-09105, Phone: (370-5) 2127472, Fax: (370-5) 276 1380, Email: info@teltonika.

VRT012 User s guide V0.1. Address: Žirmūnų g. 27, Vilnius LT-09105, Phone: (370-5) 2127472, Fax: (370-5) 276 1380, Email: info@teltonika. VRT012 User s gude V0.1 Thank you for purchasng our product. We hope ths user-frendly devce wll be helpful n realsng your deas and brngng comfort to your lfe. Please take few mnutes to read ths manual

More information

Level Annuities with Payments Less Frequent than Each Interest Period

Level Annuities with Payments Less Frequent than Each Interest Period Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due Symoblc approach

More information

Chapter 31B - Transient Currents and Inductance

Chapter 31B - Transient Currents and Inductance Chapter 31B - Transent Currents and Inductance A PowerPont Presentaton by Paul E. Tppens, Professor of Physcs Southern Polytechnc State Unversty 007 Objectves: After completng ths module, you should be

More information

L10: Linear discriminants analysis

L10: Linear discriminants analysis L0: Lnear dscrmnants analyss Lnear dscrmnant analyss, two classes Lnear dscrmnant analyss, C classes LDA vs. PCA Lmtatons of LDA Varants of LDA Other dmensonalty reducton methods CSCE 666 Pattern Analyss

More information

DEFINING %COMPLETE IN MICROSOFT PROJECT

DEFINING %COMPLETE IN MICROSOFT PROJECT CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMI-SP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,

More information

The Bridge Rectifier

The Bridge Rectifier 9/4/004 The Brdge ectfer.doc 1/9 The Brdge ectfer Now consder ths juncton dode rectfer crcut: 1 Lne (t) - O (t) _ 4 3 We call ths crcut the brdge rectfer. Let s analyze t and see what t does! Frst, we

More information

J. Parallel Distrib. Comput.

J. Parallel Distrib. Comput. J. Parallel Dstrb. Comput. 71 (2011) 62 76 Contents lsts avalable at ScenceDrect J. Parallel Dstrb. Comput. journal homepage: www.elsever.com/locate/jpdc Optmzng server placement n dstrbuted systems n

More information

( ) B. Application of Phasors to Electrical Networks In an electrical network, let the instantaneous voltage and the instantaneous current be

( ) B. Application of Phasors to Electrical Networks In an electrical network, let the instantaneous voltage and the instantaneous current be World Academy of Scence Engneerng and echnology Internatonal Journal of Electrcal obotcs Electroncs and ommuncatons Engneerng Vol:8 No:7 4 Analyss of Electrcal Networks Usng Phasors: A Bond Graph Approach

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

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

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

) of the Cell class is created containing information about events associated with the cell. Events are added to the Cell instance

) of the Cell class is created containing information about events associated with the cell. Events are added to the Cell instance Calbraton Method Instances of the Cell class (one nstance for each FMS cell) contan ADC raw data and methods assocated wth each partcular FMS cell. The calbraton method ncludes event selecton (Class Cell

More information

Simple Interest Loans (Section 5.1) :

Simple Interest Loans (Section 5.1) : Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part

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

Laddered Multilevel DC/AC Inverters used in Solar Panel Energy Systems

Laddered Multilevel DC/AC Inverters used in Solar Panel Energy Systems Proceedngs of the nd Internatonal Conference on Computer Scence and Electroncs Engneerng (ICCSEE 03) Laddered Multlevel DC/AC Inverters used n Solar Panel Energy Systems Fang Ln Luo, Senor Member IEEE

More information

Minimal Coding Network With Combinatorial Structure For Instantaneous Recovery From Edge Failures

Minimal Coding Network With Combinatorial Structure For Instantaneous Recovery From Edge Failures Mnmal Codng Network Wth Combnatoral Structure For Instantaneous Recovery From Edge Falures Ashly Joseph 1, Mr.M.Sadsh Sendl 2, Dr.S.Karthk 3 1 Fnal Year ME CSE Student Department of Computer Scence Engneerng

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

The Development of Web Log Mining Based on Improve-K-Means Clustering Analysis

The Development of Web Log Mining Based on Improve-K-Means Clustering Analysis The Development of Web Log Mnng Based on Improve-K-Means Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.

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

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

We assume your students are learning about self-regulation (how to change how alert they feel) through the Alert Program with its three stages:

We assume your students are learning about self-regulation (how to change how alert they feel) through the Alert Program with its three stages: Welcome to ALERT BINGO, a fun-flled and educatonal way to learn the fve ways to change engnes levels (Put somethng n your Mouth, Move, Touch, Look, and Lsten) as descrbed n the How Does Your Engne Run?

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

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

Optical Signal-to-Noise Ratio and the Q-Factor in Fiber-Optic Communication Systems

Optical Signal-to-Noise Ratio and the Q-Factor in Fiber-Optic Communication Systems Applcaton ote: FA-9.0. Re.; 04/08 Optcal Sgnal-to-ose Rato and the Q-Factor n Fber-Optc Communcaton Systems Functonal Dagrams Pn Confguratons appear at end of data sheet. Functonal Dagrams contnued at

More information

In our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount in the account, the balance, is

In our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount in the account, the balance, is Payout annutes: Start wth P dollars, e.g., P = 100, 000. Over a 30 year perod you receve equal payments of A dollars at the end of each month. The amount of money left n the account, the balance, earns

More information

Compiling for Parallelism & Locality. Dependence Testing in General. Algorithms for Solving the Dependence Problem. Dependence Testing

Compiling for Parallelism & Locality. Dependence Testing in General. Algorithms for Solving the Dependence Problem. Dependence Testing Complng for Parallelsm & Localty Dependence Testng n General Assgnments Deadlne for proect 4 extended to Dec 1 Last tme Data dependences and loops Today Fnsh data dependence analyss for loops General code

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

Can Auto Liability Insurance Purchases Signal Risk Attitude?

Can Auto Liability Insurance Purchases Signal Risk Attitude? Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159-164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? Chu-Shu L Department of Internatonal Busness, Asa Unversty, Tawan Sheng-Chang

More information

Forecasting the Direction and Strength of Stock Market Movement

Forecasting the Direction and Strength of Stock Market Movement Forecastng the Drecton and Strength of Stock Market Movement Jngwe Chen Mng Chen Nan Ye cjngwe@stanford.edu mchen5@stanford.edu nanye@stanford.edu Abstract - Stock market s one of the most complcated systems

More information

Hollinger Canadian Publishing Holdings Co. ( HCPH ) proceeding under the Companies Creditors Arrangement Act ( CCAA )

Hollinger Canadian Publishing Holdings Co. ( HCPH ) proceeding under the Companies Creditors Arrangement Act ( CCAA ) February 17, 2011 Andrew J. Hatnay ahatnay@kmlaw.ca Dear Sr/Madam: Re: Re: Hollnger Canadan Publshng Holdngs Co. ( HCPH ) proceedng under the Companes Credtors Arrangement Act ( CCAA ) Update on CCAA Proceedngs

More information

1.1 The University may award Higher Doctorate degrees as specified from time-to-time in UPR AS11 1.

1.1 The University may award Higher Doctorate degrees as specified from time-to-time in UPR AS11 1. HIGHER DOCTORATE DEGREES SUMMARY OF PRINCIPAL CHANGES General changes None Secton 3.2 Refer to text (Amendments to verson 03.0, UPR AS02 are shown n talcs.) 1 INTRODUCTION 1.1 The Unversty may award Hgher

More information

Single and multiple stage classifiers implementing logistic discrimination

Single and multiple stage classifiers implementing logistic discrimination Sngle and multple stage classfers mplementng logstc dscrmnaton Hélo Radke Bttencourt 1 Dens Alter de Olvera Moraes 2 Vctor Haertel 2 1 Pontfíca Unversdade Católca do Ro Grande do Sul - PUCRS Av. Ipranga,

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

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

On the Solution of Indefinite Systems Arising in Nonlinear Optimization

On the Solution of Indefinite Systems Arising in Nonlinear Optimization On the Soluton of Indefnte Systems Arsng n Nonlnear Optmzaton Slva Bonettn, Valera Ruggero and Federca Tnt Dpartmento d Matematca, Unverstà d Ferrara Abstract We consder the applcaton of the precondtoned

More information

Inter-Ing 2007. INTERDISCIPLINARITY IN ENGINEERING SCIENTIFIC INTERNATIONAL CONFERENCE, TG. MUREŞ ROMÂNIA, 15-16 November 2007.

Inter-Ing 2007. INTERDISCIPLINARITY IN ENGINEERING SCIENTIFIC INTERNATIONAL CONFERENCE, TG. MUREŞ ROMÂNIA, 15-16 November 2007. Inter-Ing 2007 INTERDISCIPLINARITY IN ENGINEERING SCIENTIFIC INTERNATIONAL CONFERENCE, TG. MUREŞ ROMÂNIA, 15-16 November 2007. UNCERTAINTY REGION SIMULATION FOR A SERIAL ROBOT STRUCTURE MARIUS SEBASTIAN

More information

SUPPLIER FINANCING AND STOCK MANAGEMENT. A JOINT VIEW.

SUPPLIER FINANCING AND STOCK MANAGEMENT. A JOINT VIEW. SUPPLIER FINANCING AND STOCK MANAGEMENT. A JOINT VIEW. Lucía Isabel García Cebrán Departamento de Economía y Dreccón de Empresas Unversdad de Zaragoza Gran Vía, 2 50.005 Zaragoza (Span) Phone: 976-76-10-00

More information

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange

More information

Enriching the Knowledge Sources Used in a Maximum Entropy Part-of-Speech Tagger

Enriching the Knowledge Sources Used in a Maximum Entropy Part-of-Speech Tagger Enrchng the Knowledge Sources Used n a Maxmum Entropy Part-of-Speech Tagger Krstna Toutanova Dept of Computer Scence Gates Bldg 4A, 353 Serra Mall Stanford, CA 94305 9040, USA krstna@cs.stanford.edu Chrstopher

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

Hosted Voice Self Service Installation Guide

Hosted Voice Self Service Installation Guide Hosted Voce Self Servce Installaton Gude Contact us at 1-877-355-1501 learnmore@elnk.com www.earthlnk.com 2015 EarthLnk. Trademarks are property of ther respectve owners. All rghts reserved. 1071-07629

More information

The program for the Bachelor degrees shall extend over three years of full-time study or the parttime equivalent.

The program for the Bachelor degrees shall extend over three years of full-time study or the parttime equivalent. Bachel of Commerce Bachel of Commerce (Accountng) Bachel of Commerce (Cpate Fnance) Bachel of Commerce (Internatonal Busness) Bachel of Commerce (Management) Bachel of Commerce (Marketng) These Program

More information

Logical Development Of Vogel s Approximation Method (LD-VAM): An Approach To Find Basic Feasible Solution Of Transportation Problem

Logical Development Of Vogel s Approximation Method (LD-VAM): An Approach To Find Basic Feasible Solution Of Transportation Problem INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME, ISSUE, FEBRUARY ISSN 77-866 Logcal Development Of Vogel s Approxmaton Method (LD- An Approach To Fnd Basc Feasble Soluton Of Transportaton

More information

Financial Mathemetics

Financial Mathemetics Fnancal Mathemetcs 15 Mathematcs Grade 12 Teacher Gude Fnancal Maths Seres Overvew In ths seres we am to show how Mathematcs can be used to support personal fnancal decsons. In ths seres we jon Tebogo,

More information

University Physics AI No. 11 Kinetic Theory

University Physics AI No. 11 Kinetic Theory Unersty hyscs AI No. 11 Knetc heory Class Number Name I.Choose the Correct Answer 1. Whch type o deal gas wll hae the largest alue or C -C? ( D (A Monatomc (B Datomc (C olyatomc (D he alue wll be the same

More information