The modelling of business rules for dashboard reporting using mutual information

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "The modelling of business rules for dashboard reporting using mutual information"

Transcription

1 8 t World IMACS / MODSIM Congress, Cairns, Australia 3-7 July 2009 ttp://mssanz.org.au/modsim09 Te modelling of business rules for dasboard reporting using mutual information Gregory Calbert Command, Control, Communications and Intelligence Division, Defence Science and Tecnology Organisation, Edinburg, Sout Australia, 5. Abstract: Te role of a business or military dasboard is to form a succinct picture of te key performance indicators tat govern an organisation s dynamics or effectiveness. Tese indicators are based on te judgments of different experts. Variables from one or several databases along wit te subjective opinion of te expert are fused or amalgamated to form te so called key performance indicators. A principal example of suc a dasboard is te reporting tat occurs in te defence preparedness system. Judgments, based on data or opinions witin lower level units are formed ten fused troug te defence ierarcy. Te aim of tis system is to form a concise picture of te strategically important preparedness issues pertinent to senior leadersip. Te issue tat we begin to address in tis paper is te analysis of just ow informative suc business dasboards can be. Tere are tree principal reasons as to wy te information conveyed by te dasboard to senior boards or defence leadersip may be uninformative. Te first is tat te selection of te key performance variables may not capture te true dynamics of te organisation s state over time. Te second is tat te models or algoritms used to map fundamental database variables onto key performance indicators may be wrong. Finally, even if te models are correct, te key performance indicator may not convey sufficient information as to te state of te organisation. In tis paper we will discuss te last issue-tat of te information conveyed troug te application of business rules to form te key performance indicator. A number of examples will suffice to illustrate te loss of information. Important information migt be lost wen variables are fused or amalgamated due to te requirement for brevity. Take for example overall profit. Wile te profit for an organisation or division may be good, te successes of some brances may ide a critical loss in oters. A simpler example is te requirement for ierarcical reporting. Reports are amalgamated at te branc level and ten to te division level wit inevitable loss of information. For tis paper we apply te information-teoretic concept of mutual information between te performance indicators -te state- at one level of te organisational ierarcy and te corresponding state, formed by te business rule for fusion at a iger level. Te analysis is done for a number of business rules. Te business rules analysed are te commonly applied report by exception rule or te report by majority rule. By calculation of te mutual information, one is able to quantify just ow muc information is lost troug te application of suc rules, as reports propagate upward in te ierarcy. We draw some conclusions as to wic business rule is more appropriate in organisations tat are eiter relatively static wit few key performance indicators canges, versus organisations wic are igly dynamic, were suc indicators may cange often. A generalised business rule is ten defined. Tis work forms te basis for te measurement of te effectiveness of te reporting system itself. Measures of system bias are formed tat suggest te degree of effectiveness of te overall system in te communication of te defence system s overall state to te apex of te ierarcy, tat is to senior leadersip. Keywords: Organisational modelling, information teory, ierarcy, business rule, fusion, tresold. 594

2 . INTRODUCTION Te defence preparedness system informs senior leadersip as to te overall state of te organisation in relation to te readiness of units and te ability to conduct specific operations. Reports regarding te state of a number of independent key performance indicators (KPI) are formed at lower levels of te ierarcy, typically, if using te Army as an example, at te company level. Tese reports may be in regard to te state of te facilities or stocks at different barracks. Toug tere may be some minimal dependence on te state of KPI at te different levels of te ierarcy, for pragmatic reasons tey are assumed to be independent. Tey are ten fused at te battalion, ten te brigade and finally te output group level to form a picture, termed in business a dasboard, to inform senior leadersip. Brevity and clarity is paramount and terefore te key performance indicators are drawn from a discrete set. An example of suc a set is te traffic ligt indicator form of green for adequate performance and yellow, red for moderate and inadequate performance respectively. For simplicity of analysis we will only consider te binary valued green and red KPI states of adequate and inadequate performance. As illustrated in Figure, independent reports at te lower level of te ierarcy are amalgamated troug te application of a business rule to form an overall judgement at te next level, of te ierarcy. Tis process is repeated to te apex of te ierarcy or to some dept at a iger level. Te aim of tis paper is to quantify te performance of te business rules used to form te judgments. We use an information teoretic concept of mutual-information (Sannon, 948) to quantify suc performance. A discussion as to te use of tis measure against oter measures will be made. It sould be empasised tat te application of mutual information in control and to understand networks is not new. Indeed te communications teory subject of rate distortion is devoted to te study of information loss under various compression algoritms. Tere is an extensive literature on information teory and data fusion (Tay, 2008). Recently, mutual information as been applied to quantifying information propagation in general complex networks (Luque, 997), neurobiological circuits and networks of gene dependency in biology (Butte, 2000). Our application ere is in te specific context of defence preparedness in order to draw conclusions regarding te different business rules applied. 2. MEASURES OF EFFECTIVENESS Before proceeding, it is wort defining te notation tat will be used to model te information propagation in te defence preparedness system. We define te state of a single node in te preparedness ierarcy (tat is a unit suc as a company lower in te ierarcy or a brigade or output group near te apex) to be X ( R, G), were R is te state red and G is te state green for inadequate and adequate performance respectively. Te levels of te reporting ierarcy are labeled from lowest to igest as n = 0,,, N. A combined state at level n of te ierarcy (a series of independent red and green reports) Xn = ( X n, X n2,, X nm ) is associated wit te coordinator or fuser at te next level n +. Here m is te number of performance indicators linked to te fusion state in te next level of te ierarcy, te brancing factor. Witout loss of generality we label te combined state along wit te fusion state as X, Y respectively as is sown in Figure. Te state X is termed te subordinate state to Y. Business rules are defined to be Boolean functions tat map te subordinate level combined state onto te fusion state, B : X Y. Te space of all business rules is termed Β. Tus te business rule is defined between te subordinate states to form te fused state of te coordinator. Y X B Figure. Scematic representation of te defence preparedness system, in wic te independent lower level states labelled as X are fused troug a business rule B to te state Y, te states being good (green) or bad (red). 595

3 Now we turn our attention to a discussion of te measures of performance for te preparedness system. One could argue tat tere are two broad measures. One is to propagate as many reports of inadequate performance troug to te next level of te ierarcy as possible. Framed as an optimisation problem, we look for a business rule B tat maximises te probability of reporting a red state troug to te next level of te ierarcy max Pr(red reported in level Y red in X ). B Β It is clear tat te business rule termed report by exception maximises effectiveness according to tis criterion. Te report by exception rule sees te coordinator report a red at a node if any one of subordinate nodes is red. Tis is te commonly used business rule applied in defence preparedness reporting. Anoter approac is to find a business rule tat reflects, troug canges in te fused state Y te canging states at te subordinate level X. Intuitively, consider two organisations, one were te system is stable and performance is generally good. Here, red reports are rare. Reports to te apex of a multi-level ierarcy will contain a mix of a few inadequately performing divisions, amongst te sound performers. Now consider an organisation in a state of cange or urgency. Here, tere are many units, brances or divisions wit inadequate performance. Applying te report by exception rule will mean tat over te levels, te confluence of reports at te apex of te ierarcy will only see inadequate performance across all te divisions, witout viewing te canging dynamic of te nodes below. In te limit, tere need only be one report of red for te wole system to record inadequate performance as a wole (if all nodes including te apex apply te exception rule), wic will not be reflective of te performance of all te nodes below. To empasise tis point it is wort forming a novel analogy. If all traffic ligts witin a metropolitan traffic system were red all te time, teir effectiveness in te control of traffic would cease as tey ultimately would be ignored. To prevent tis problem of reporting in dynamic and canging organisations we propose using te mutual information between levels of te ierarcy as te measure of effectiveness. Suppose tat te evolution of te subordinate state over time is formed from a probability distribution Pr(X ). Te measure of te information (in bits) of te subordinate state is te information entropy H( X) = p ( X)log p ( X ). X Te entropy is a measure of ow muc information te random variable X conveys over time (Sannon, 948). Between te subordinate level X and te fusion state Y te mutual information is defined as I( X, Y) = H( X) H( X Y). Tis function is symmetric and measures ow te dynamic canges of X over time are reflected in Y. We can observe tis noting tat if X and Y are independent ten H( X Y) = H( X) making te mutual information zero. If X as te same distribution as Y ten H( X Y) = H( X X) = 0 so te mutual information is maximised to H( X). We tus seek a business rule or series of business rules tat maximise te mutual information between levels of te ierarcy, as is seen from te commonly formed capacity equation (Sannon, 948) max I( X,Y). B Β 3. THE MUTUAL INFORMATION OF TWO BUSINESS RULES We will derive te mutual information of two business rules in tis section. Te first is te aforementioned report by exception rule. We ten consider te report by majority rule. For tis rule, te reporter in te fusion node above observes all subordinate node states. If tere are more green states tan red ten a report of green is made, oterwise red is reported. Bot rules are juxtaposed in Figure 2. Of course tere will be many oter rules tat may be applicable. For example tere may be special subordinate nodes tat ave more weigt in te reporting tan oters. We do not consider tis, as all nodes are considered equally important. More important nodes may be expanded to multiple nodes wit more detailed representation. Generally, we consider tresold business rules (Irving, 994), were, exceeding some specified fraction of red reports te lowest common ancestor reports red

4 Our aim is to examine ow tese two business rules function in conveying information to te next level of te ierarcy, under different levels of organisational uncertainty. Drawing from te examples of Section 2, organisations tat face certain environments ave low entropy wilst ig uncertainty implies rapid canges over time in te value of te key performance indicators, tat is, ig entropy. To calculate te mutual information for te two rules, a model of te key performance indicator values (te value of te nodes) must be formed at level 0 of te ierarcy (te base nodes)-te interface between te organisation and te environment. Our approac is to define a set of distributions parameterised by entropy value in te following way. Assume tere are m subordinate nodes tat report to te coordinator or fuser. Suppose X 0 = ( X 0, X 02,, X 0m ) ten in order to tune te distribution over X0 to ave entropy te first m nodes are set or frozen to te green state, X 0 = X02 = = X0 ( m ) = G, wit probability. Te remaining nodes are set to ave eiter te green or red state wit probability / 2. Tus Pr ( X ) = Pr( X ) Pr( X ) Pr( X ), Pr( X = G) = / 2, i m +,, m. o m + m +2 m i It is easy to sow tat te total entropy of te subordinate nodes defined by tis distribution as entropy bits. Te base nodes tat can switc from red or green states wit probability / 2 supply te uncertainty into te ierarcy. Tus te entropy, is bot te measure of te uncertainty of te organisation and te number of subordinate nodes tat can switc between te red and green states. Wit tis explicit model distribution, we are able to calculate te mutual information precisely for te two business rules. Te examination of te effectiveness of eac rule can ten be ascertained by varying te entropy, tat is canging te number of nodes tat can switc between te red and green states. To calculate te mutual information te conditional entropy between te fusion node states and te subordinate node states H( X Y) Pr( Y) Pr( X Y)log Pr( X Y ) = Y=R, G X is evaluated. We omit te details of te derivation albeit to say tat te conditional distributions are evaluated troug te application of Baye s Teorem, Pr( X Y) = (Pr( Y X) Pr( X)) / Pr( Y). Upon viewing te conditional probabilities, te two business rules can now be expressed in general form. For bot rules te conditional probability were for te report by exception rule 2 for X S Pr( Y = R X) = R 0,oterwise, EXCEPTION { X } S R,exception = : te number of red subordinate reports and for te report by te majority rule wit m subordinates reporting to te coordinator { X m } SR,majority = : te number of red subordinate reports / 2. MAJORITY Figure 2. Illustration of te exception and minority reporting rules. On te left and side a red state is seen so under te exception rules a red is reported for te fusion. On te rigt and side tere are more green reports tan red, so under te majority rule a green is reported. Put in tis form, te reporting rule can be generalised to any tresold rule as we will discuss in te following section 4. Te mutual information between te subordinate nodes and te coordinator node for te exception rule, given information entropy level of te subordinate nodes and te distribution specified above can be sown to be 597

5 ( ) I exception ( X, Y ) = log2 2, >0. 2 Te report by majority rule can be sown to ave mutual information K 2 log2( 2 K) + log2 I majority ( ) X, Y = 2 K for > m/2, 0oterwise, were te number of combined subordinate states wit total entropy in wic te majority report is red (tat is in S ) R,majority K = m/2 m/2 + Te following figure plots te mutual information for te exception and te majority rules, as a function of te information entropy, assuming tat tere are eleven suc subordinate nodes. Mutual Information exception majority Total Subordinate Entropy (Nodes tat can switc states) Figure 3: Mutual information between subordinate and fusion nodes of te exception and majority reporting rules as a function of te subordinate nodes total information entropy, wic is te number of nodes tat can switc between red and green states, given eleven subordinate nodes. It is clear from inspection of Figure 3 tat te exception reporting rule is most effective wen te information entropy is low. Te interpretation for te preparedness reporting system is tat te exception rule applies wen most subordinate reports are consistently good (green) over time, reporting few inadequate key performance indicators. However, as te uncertainty increases te majority reporting rule ten becomes more informative as to te state of te subordinate nodes below. Analogously, tis reflects an organisation were te base key performance indicators are in flux and cange over time. Referring to Figure 3, in te left limit as te information entropy approaces one, any canges in te subordinate states are precisely reflected in te report of te lowest common ancestor, as tere is one subordinate state canging. Hence te mutual information is maximised for te exception rule. Conversely, in te rigt limit, wen all subordinate states can cange randomly and equiprobably, te majority rule will reflect tese canges, toug not precisely as tere is information loss. Te mutual information approaces te maximum of one (tis will be te maximum for te coordinator node as it can only be in two states, red and green, one bit). Te mutual information reaces tis maximum value only because our assumption tat all subordinate nodes can cange in an equiprobable way. If nodes can cange from across red and green states wit different probabilities ten te mutual information for te majority rule will not reac one. We discuss tis issue in te following section. 598

6 4. A GENERALISED THRESHOLD BUSINESS RULE Troug te preceding section we ave illustrated tat te exception business rule may not necessarily be effective in propagating te canging states of te subordinate nodes below. We modeled te majority business rule and found tat under te circumstances of dynamic state cange of subordinate nodes, tis rule better reflects suc a cange in states. Bot euristics are a version of te general tresold business rule as determined by te value of te conditional probability distribution Pr( Y = R X) wic is one (tat is te business rule reports te red state) under our distribution Pr ( X 0) for te set { X a} SRa, = : te number of red subordinate reports. For te exception rule a = and for te majority rule a = m / 2 were m is te number of subordinate nodes. It is natural to ask if tere is an generalised tresold business rule. Couced in communications teory terms we seek a rule maximises tat capacity between te subordinate nodes and te state of te fusion node. Under our distribution Pr ( X0) defined earlier, we can calculate te mutual information for a general value of a. Te resulting equation is identical to tat of te mutual information for te majority rule except tat te value of K is replaced by a term dependent on a, tis being te number of elements in SRa, wit entropy bits K( a) = a a+ One need only cycle over te values following Figure 4. 0 a H to find te optimum value of a. Tese values are sown in te a, te number of subordinate red states before reporting red Total Subordinate Entropy (Number of nodes at can switc states) Figure 4: Grap of te optimal number of red states a, given eleven subordinate nodes, before te business rule reports an overall state of red, as a function of te subordinate entropy. Referring to Figure 4, as per te observation in Figure 3, as te uncertainty in te organisation increases, te business rule tresold by wic te subordinate reports te red state also increases. As an approximation, a = /2. te tresold number of red reports [ ] We are able to calculate a generalised business rule under te assumption tat all te nodes tat can switc between red and green states do so wit equal probability / 2 over time. If tis assumption fails, ten inevitably, te difficulty of calculating exact expressions for te mutual information will increase substantially. To overcome tis problem, one need only employ Monte Carlo simulation for te given set of state switcing probabilities to determine wat te tresold will be. For tat set of state switcing probabilities Pr( = G, R), i =, m te information entropy may be exactly determined. Tis gives us an X i 599

7 upper bound on a as 0 a. It is only te conditional probabilities tat need to be estimated from simulation. 5. ESTIMATES OF BIAS IN THE REPORTING SYSTEM From te discussion above, we are able to establis criteria by wic reporting witin te preparedness system is biased. Tis measure of bias will determine te overall system effectiveness. Longitudinal data on te state canges of eac node in te current defence system is recorded and terefore te entropy of all subordinate nodes can be estimated. Tere are two approaces to estimating te internal preparedness system bias, given longitudinal data. One is in reference to te judgments of oter reports in te ierarcy and te oter is in reference to te deviation from te generalised business rule or te mandated business rule. To measure te internal bias in reporting, one can estimate, given a particular level of te ierarcy te probability of a fused state Pr( Y X). Differences in te fused state probabilities can be calculated and summed to form te bias estimate over te wole ierarcy. Assuming tat te subordinate states are te same, tat is X = X for two fused states Yi and Yj, a measure of te preparedness system bias is Yi Yj b = i Yi j Yj Hierarcylevels i, j Pr( Y X ) Pr( Y X ). An alternative is to look at te deviation from te result of te general business rule. For a fusion node, an indicator function may be defined * (Y) tat is one wen te fused state matces tat of te state found Y from te business rule or zero oterwise. Ten one measure of te preparedness system bias, given te set of fusion nodes wic we label A (tat is all nodes except te base nodes of te ierarcy) is 6. DISCUSSION AND CONCLUSION ( ) b= A *. Y Y i i i A We ave igligted tat te use of te report by exception rule, wilst appropriate for a preparedness system tat as few problems, may not be te appropriate rule wen te values of te key performance indicators sow dynamic canges over time. Te majority business rule or te general tresold rule as discussed in Section 4 is more appropriate. In Defence, decision makers know te problems wit te exception rule. At te moment tere is considerable subjectivity and variability in fusion decisions made, wit limited guidance as to wat rules sould be applied across different circumstances. Decision makers ave te option to fuse decisions based on exception or averaged results from te subordinates or fuse subjectively. Data regarding te longitudinal cange in states over time is recorded. As suc, te necessary infrastructure is in place at present to apply te best tresold business rules as derived in tis paper. Tis work sould improve te guidance on wat rules to apply to best relay preparedness information to senior leadersip. REFERENCES Butte, A. J. and Koane, I.S. (2000), Mutual Information Relevance Networks: Functional Genomic Clustering Using Pairwise Entropy Estimates. Pacific Symposium in Biocomputing, Irving, W. W. and Tsitsiklis, J. N., (994), Some Properties of Optimal Tresolds in Decentralized Detection. IEEE Transactions in Automatic Control 39(4). Luque, B. and Ferrera, A., (997), Measuring Mutual Information in Random Boolean Networks. Complex Systems,. Sannon, C.E. (948), A Matematical Teory of Communication. Te Bell System Tecnical Journal and Tay, W. P., Tsitsiklis, J. N., and Win, M. Z., (2008), Data Fusion Trees for Detection: Does Arcitecture Matter? IEEE Transactions in Information Teory 54(9),

Understanding the Derivative Backward and Forward by Dave Slomer

Understanding the Derivative Backward and Forward by Dave Slomer Understanding te Derivative Backward and Forward by Dave Slomer Slopes of lines are important, giving average rates of cange. Slopes of curves are even more important, giving instantaneous rates of cange.

More information

Geometric Stratification of Accounting Data

Geometric Stratification of Accounting Data Stratification of Accounting Data Patricia Gunning * Jane Mary Horgan ** William Yancey *** Abstract: We suggest a new procedure for defining te boundaries of te strata in igly skewed populations, usual

More information

Optimal Pricing Strategy for Second Degree Price Discrimination

Optimal Pricing Strategy for Second Degree Price Discrimination Optimal Pricing Strategy for Second Degree Price Discrimination Alex O Brien May 5, 2005 Abstract Second Degree price discrimination is a coupon strategy tat allows all consumers access to te coupon. Purcases

More information

Finite Difference Approximations

Finite Difference Approximations Capter Finite Difference Approximations Our goal is to approximate solutions to differential equations, i.e., to find a function (or some discrete approximation to tis function) tat satisfies a given relationsip

More information

Derivatives Math 120 Calculus I D Joyce, Fall 2013

Derivatives Math 120 Calculus I D Joyce, Fall 2013 Derivatives Mat 20 Calculus I D Joyce, Fall 203 Since we ave a good understanding of its, we can develop derivatives very quickly. Recall tat we defined te derivative f x of a function f at x to be te

More information

2.0 5-Minute Review: Polynomial Functions

2.0 5-Minute Review: Polynomial Functions mat 3 day 3: intro to limits 5-Minute Review: Polynomial Functions You sould be familiar wit polynomials Tey are among te simplest of functions DEFINITION A polynomial is a function of te form y = p(x)

More information

Math 113 HW #5 Solutions

Math 113 HW #5 Solutions Mat 3 HW #5 Solutions. Exercise.5.6. Suppose f is continuous on [, 5] and te only solutions of te equation f(x) = 6 are x = and x =. If f() = 8, explain wy f(3) > 6. Answer: Suppose we ad tat f(3) 6. Ten

More information

7.6 Complex Fractions

7.6 Complex Fractions Section 7.6 Comple Fractions 695 7.6 Comple Fractions In tis section we learn ow to simplify wat are called comple fractions, an eample of wic follows. 2 + 3 Note tat bot te numerator and denominator are

More information

The differential amplifier

The differential amplifier DiffAmp.doc 1 Te differential amplifier Te emitter coupled differential amplifier output is V o = A d V d + A c V C Were V d = V 1 V 2 and V C = (V 1 + V 2 ) / 2 In te ideal differential amplifier A c

More information

M(0) = 1 M(1) = 2 M(h) = M(h 1) + M(h 2) + 1 (h > 1)

M(0) = 1 M(1) = 2 M(h) = M(h 1) + M(h 2) + 1 (h > 1) Insertion and Deletion in VL Trees Submitted in Partial Fulfillment of te Requirements for Dr. Eric Kaltofen s 66621: nalysis of lgoritms by Robert McCloskey December 14, 1984 1 ackground ccording to Knut

More information

This supplement is meant to be read after Venema s Section 9.2. Throughout this section, we assume all nine axioms of Euclidean geometry.

This supplement is meant to be read after Venema s Section 9.2. Throughout this section, we assume all nine axioms of Euclidean geometry. Mat 444/445 Geometry for Teacers Summer 2008 Supplement : Similar Triangles Tis supplement is meant to be read after Venema s Section 9.2. Trougout tis section, we assume all nine axioms of uclidean geometry.

More information

Trapezoid Rule. y 2. y L

Trapezoid Rule. y 2. y L Trapezoid Rule and Simpson s Rule c 2002, 2008, 200 Donald Kreider and Dwigt Lar Trapezoid Rule Many applications of calculus involve definite integrals. If we can find an antiderivative for te integrand,

More information

Verifying Numerical Convergence Rates

Verifying Numerical Convergence Rates 1 Order of accuracy Verifying Numerical Convergence Rates We consider a numerical approximation of an exact value u. Te approximation depends on a small parameter, suc as te grid size or time step, and

More information

Instantaneous Rate of Change:

Instantaneous Rate of Change: Instantaneous Rate of Cange: Last section we discovered tat te average rate of cange in F(x) can also be interpreted as te slope of a scant line. Te average rate of cange involves te cange in F(x) over

More information

An Interest Rate Model

An Interest Rate Model An Interest Rate Model Concepts and Buzzwords Building Price Tree from Rate Tree Lognormal Interest Rate Model Nonnegativity Volatility and te Level Effect Readings Tuckman, capters 11 and 12. Lognormal

More information

The EOQ Inventory Formula

The EOQ Inventory Formula Te EOQ Inventory Formula James M. Cargal Matematics Department Troy University Montgomery Campus A basic problem for businesses and manufacturers is, wen ordering supplies, to determine wat quantity of

More information

Lecture 10: What is a Function, definition, piecewise defined functions, difference quotient, domain of a function

Lecture 10: What is a Function, definition, piecewise defined functions, difference quotient, domain of a function Lecture 10: Wat is a Function, definition, piecewise defined functions, difference quotient, domain of a function A function arises wen one quantity depends on anoter. Many everyday relationsips between

More information

VOL. 6, NO. 9, SEPTEMBER 2011 ISSN ARPN Journal of Engineering and Applied Sciences

VOL. 6, NO. 9, SEPTEMBER 2011 ISSN ARPN Journal of Engineering and Applied Sciences VOL. 6, NO. 9, SEPTEMBER 0 ISSN 89-6608 006-0 Asian Researc Publising Network (ARPN). All rigts reserved. BIT ERROR RATE, PERFORMANCE ANALYSIS AND COMPARISION OF M x N EQUALIZER BASED MAXIMUM LIKELIHOOD

More information

Differentiable Functions

Differentiable Functions Capter 8 Differentiable Functions A differentiable function is a function tat can be approximated locally by a linear function. 8.. Te derivative Definition 8.. Suppose tat f : (a, b) R and a < c < b.

More information

2.28 EDGE Program. Introduction

2.28 EDGE Program. Introduction Introduction Te Economic Diversification and Growt Enterprises Act became effective on 1 January 1995. Te creation of tis Act was to encourage new businesses to start or expand in Newfoundland and Labrador.

More information

Catalogue no. 12-001-XIE. Survey Methodology. December 2004

Catalogue no. 12-001-XIE. Survey Methodology. December 2004 Catalogue no. 1-001-XIE Survey Metodology December 004 How to obtain more information Specific inquiries about tis product and related statistics or services sould be directed to: Business Survey Metods

More information

Pressure. Pressure. Atmospheric pressure. Conceptual example 1: Blood pressure. Pressure is force per unit area:

Pressure. Pressure. Atmospheric pressure. Conceptual example 1: Blood pressure. Pressure is force per unit area: Pressure Pressure is force per unit area: F P = A Pressure Te direction of te force exerted on an object by a fluid is toward te object and perpendicular to its surface. At a microscopic level, te force

More information

Solution Derivations for Capa #7

Solution Derivations for Capa #7 Solution Derivations for Capa #7 1) Consider te beavior of te circuit, wen various values increase or decrease. (Select I-increases, D-decreases, If te first is I and te rest D, enter IDDDD). A) If R1

More information

f(a + h) f(a) f (a) = lim

f(a + h) f(a) f (a) = lim Lecture 7 : Derivative AS a Function In te previous section we defined te derivative of a function f at a number a (wen te function f is defined in an open interval containing a) to be f (a) 0 f(a + )

More information

Investigating Cost-Efficient Ways of Running a Survey of Pacific Peoples. Undertaken for the Ministry of Pacific Island Affairs

Investigating Cost-Efficient Ways of Running a Survey of Pacific Peoples. Undertaken for the Ministry of Pacific Island Affairs Investigating Cost-Efficient Ways of Running a Survey of Pacific Peoples Undertaken for te Ministry of Pacific Island Affairs by Temaleti Tupou, assisted by Debra Taylor and Tracey Savage Survey Metods,

More information

1 Density functions, cummulative density functions, measures of central tendency, and measures of dispersion

1 Density functions, cummulative density functions, measures of central tendency, and measures of dispersion Density functions, cummulative density functions, measures of central tendency, and measures of dispersion densityfunctions-intro.tex October, 9 Note tat tis section of notes is limitied to te consideration

More information

Note nine: Linear programming CSE 101. 1 Linear constraints and objective functions. 1.1 Introductory example. Copyright c Sanjoy Dasgupta 1

Note nine: Linear programming CSE 101. 1 Linear constraints and objective functions. 1.1 Introductory example. Copyright c Sanjoy Dasgupta 1 Copyrigt c Sanjoy Dasgupta Figure. (a) Te feasible region for a linear program wit two variables (see tet for details). (b) Contour lines of te objective function: for different values of (profit). Te

More information

Is Gravity an Entropic Force?

Is Gravity an Entropic Force? Is Gravity an Entropic Force? San Gao Unit for HPS & Centre for Time, SOPHI, University of Sydney, Sydney, NSW 006, Australia; E-Mail: sgao7319@uni.sydney.edu.au Abstract: Te remarkable connections between

More information

Reliability and Route Diversity in Wireless Networks

Reliability and Route Diversity in Wireless Networks Reliability and Route Diversity in Wireless Networks Amir E. Kandani, Jinane Abounadi, Eytan Modiano, and Lizong Zeng Abstract We study te problem of communication reliability of wireless networks in a

More information

2 Limits and Derivatives

2 Limits and Derivatives 2 Limits and Derivatives 2.7 Tangent Lines, Velocity, and Derivatives A tangent line to a circle is a line tat intersects te circle at exactly one point. We would like to take tis idea of tangent line

More information

- 1 - Handout #22 May 23, 2012 Huffman Encoding and Data Compression. CS106B Spring 2012. Handout by Julie Zelenski with minor edits by Keith Schwarz

- 1 - Handout #22 May 23, 2012 Huffman Encoding and Data Compression. CS106B Spring 2012. Handout by Julie Zelenski with minor edits by Keith Schwarz CS106B Spring 01 Handout # May 3, 01 Huffman Encoding and Data Compression Handout by Julie Zelenski wit minor edits by Keit Scwarz In te early 1980s, personal computers ad ard disks tat were no larger

More information

Proof of the Power Rule for Positive Integer Powers

Proof of the Power Rule for Positive Integer Powers Te Power Rule A function of te form f (x) = x r, were r is any real number, is a power function. From our previous work we know tat x x 2 x x x x 3 3 x x In te first two cases, te power r is a positive

More information

Finite Volume Discretization of the Heat Equation

Finite Volume Discretization of the Heat Equation Lecture Notes 3 Finite Volume Discretization of te Heat Equation We consider finite volume discretizations of te one-dimensional variable coefficient eat equation, wit Neumann boundary conditions u t x

More information

Week #15 - Word Problems & Differential Equations Section 8.2

Week #15 - Word Problems & Differential Equations Section 8.2 Week #1 - Word Problems & Differential Equations Section 8. From Calculus, Single Variable by Huges-Hallett, Gleason, McCallum et. al. Copyrigt 00 by Jon Wiley & Sons, Inc. Tis material is used by permission

More information

A.4. Rational Expressions. Domain of an Algebraic Expression. What you should learn. Why you should learn it

A.4. Rational Expressions. Domain of an Algebraic Expression. What you should learn. Why you should learn it A6 Appendi A Review of Fundamental Concepts of Algebra A.4 Rational Epressions Wat you sould learn Find domains of algebraic epressions. Simplify rational epressions. Add, subtract, multiply, and divide

More information

Hardness Measurement of Metals Static Methods

Hardness Measurement of Metals Static Methods Hardness Testing Principles and Applications Copyrigt 2011 ASM International Konrad Herrmann, editor All rigts reserved. www.asminternational.org Capter 2 Hardness Measurement of Metals Static Metods T.

More information

11.2 Instantaneous Rate of Change

11.2 Instantaneous Rate of Change 11. Instantaneous Rate of Cange Question 1: How do you estimate te instantaneous rate of cange? Question : How do you compute te instantaneous rate of cange using a limit? Te average rate of cange is useful

More information

ME422 Mechanical Control Systems Modeling Fluid Systems

ME422 Mechanical Control Systems Modeling Fluid Systems Cal Poly San Luis Obispo Mecanical Engineering ME422 Mecanical Control Systems Modeling Fluid Systems Owen/Ridgely, last update Mar 2003 Te dynamic euations for fluid flow are very similar to te dynamic

More information

Optimized Data Indexing Algorithms for OLAP Systems

Optimized Data Indexing Algorithms for OLAP Systems Database Systems Journal vol. I, no. 2/200 7 Optimized Data Indexing Algoritms for OLAP Systems Lucian BORNAZ Faculty of Cybernetics, Statistics and Economic Informatics Academy of Economic Studies, Bucarest

More information

An inquiry into the multiplier process in IS-LM model

An inquiry into the multiplier process in IS-LM model An inquiry into te multiplier process in IS-LM model Autor: Li ziran Address: Li ziran, Room 409, Building 38#, Peing University, Beijing 00.87,PRC. Pone: (86) 00-62763074 Internet Address: jefferson@water.pu.edu.cn

More information

Can a Lump-Sum Transfer Make Everyone Enjoy the Gains. from Free Trade?

Can a Lump-Sum Transfer Make Everyone Enjoy the Gains. from Free Trade? Can a Lump-Sum Transfer Make Everyone Enjoy te Gains from Free Trade? Yasukazu Icino Department of Economics, Konan University June 30, 2010 Abstract I examine lump-sum transfer rules to redistribute te

More information

Tangent Lines and Rates of Change

Tangent Lines and Rates of Change Tangent Lines and Rates of Cange 9-2-2005 Given a function y = f(x), ow do you find te slope of te tangent line to te grap at te point P(a, f(a))? (I m tinking of te tangent line as a line tat just skims

More information

P.4 Rational Expressions

P.4 Rational Expressions 7_0P04.qp /7/06 9:4 AM Page 7 Section P.4 Rational Epressions 7 P.4 Rational Epressions Domain of an Algebraic Epression Te set of real numbers for wic an algebraic epression is defined is te domain of

More information

A New Hybrid Meta-heuristic Approach for Stratified Sampling

A New Hybrid Meta-heuristic Approach for Stratified Sampling A New Hybrid Meta-euristic Approac for Stratified Sampling Timur KESKİNTÜRK 1, Sultan KUZU 2, Baadır F. YILDIRIM 3 1 Scool of Business, İstanbul University, İstanbul, Turkey 2 Scool of Business, İstanbul

More information

Section 3.3. Differentiation of Polynomials and Rational Functions. Difference Equations to Differential Equations

Section 3.3. Differentiation of Polynomials and Rational Functions. Difference Equations to Differential Equations Difference Equations to Differential Equations Section 3.3 Differentiation of Polynomials an Rational Functions In tis section we begin te task of iscovering rules for ifferentiating various classes of

More information

Differential Calculus: Differentiation (First Principles, Rules) and Sketching Graphs (Grade 12)

Differential Calculus: Differentiation (First Principles, Rules) and Sketching Graphs (Grade 12) OpenStax-CNX moule: m39313 1 Differential Calculus: Differentiation (First Principles, Rules) an Sketcing Graps (Grae 12) Free Hig Scool Science Texts Project Tis work is prouce by OpenStax-CNX an license

More information

Distributed and Centralized Bypass Architectures Compared. E. Guidotti - AC Power Product Manager A. Ferro - Key Account Manager

Distributed and Centralized Bypass Architectures Compared. E. Guidotti - AC Power Product Manager A. Ferro - Key Account Manager Distributed and Centralized Bypass Arcitectures Compared E. Guidotti - AC Power Product Manager A. Ferro - Key Account Manager Contents Executive Summary 3 Distributed / Centralized - System description

More information

Schedulability Analysis under Graph Routing in WirelessHART Networks

Schedulability Analysis under Graph Routing in WirelessHART Networks Scedulability Analysis under Grap Routing in WirelessHART Networks Abusayeed Saifulla, Dolvara Gunatilaka, Paras Tiwari, Mo Sa, Cenyang Lu, Bo Li Cengjie Wu, and Yixin Cen Department of Computer Science,

More information

Distances in random graphs with infinite mean degrees

Distances in random graphs with infinite mean degrees Distances in random graps wit infinite mean degrees Henri van den Esker, Remco van der Hofstad, Gerard Hoogiemstra and Dmitri Znamenski April 26, 2005 Abstract We study random graps wit an i.i.d. degree

More information

MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION

MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION Tis tutorial is essential pre-requisite material for anyone stuing mecanical engineering. Tis tutorial uses te principle of

More information

Sections 3.1/3.2: Introducing the Derivative/Rules of Differentiation

Sections 3.1/3.2: Introducing the Derivative/Rules of Differentiation Sections 3.1/3.2: Introucing te Derivative/Rules of Differentiation 1 Tangent Line Before looking at te erivative, refer back to Section 2.1, looking at average velocity an instantaneous velocity. Here

More information

1.6. Analyse Optimum Volume and Surface Area. Maximum Volume for a Given Surface Area. Example 1. Solution

1.6. Analyse Optimum Volume and Surface Area. Maximum Volume for a Given Surface Area. Example 1. Solution 1.6 Analyse Optimum Volume and Surface Area Estimation and oter informal metods of optimizing measures suc as surface area and volume often lead to reasonable solutions suc as te design of te tent in tis

More information

EC201 Intermediate Macroeconomics. EC201 Intermediate Macroeconomics Problem set 8 Solution

EC201 Intermediate Macroeconomics. EC201 Intermediate Macroeconomics Problem set 8 Solution EC201 Intermediate Macroeconomics EC201 Intermediate Macroeconomics Prolem set 8 Solution 1) Suppose tat te stock of mone in a given econom is given te sum of currenc and demand for current accounts tat

More information

ACT Math Facts & Formulas

ACT Math Facts & Formulas Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Rationals: fractions, tat is, anyting expressable as a ratio of integers Reals: integers plus rationals plus special numbers suc as

More information

A Low-Temperature Creep Experiment Using Common Solder

A Low-Temperature Creep Experiment Using Common Solder A Low-Temperature Creep Experiment Using Common Solder L. Roy Bunnell, Materials Science Teacer Soutridge Hig Scool Kennewick, WA 99338 roy.bunnell@ksd.org Copyrigt: Edmonds Community College 2009 Abstract:

More information

The Derivative. Not for Sale

The Derivative. Not for Sale 3 Te Te Derivative 3. Limits 3. Continuity 3.3 Rates of Cange 3. Definition of te Derivative 3.5 Grapical Differentiation Capter 3 Review Etended Application: A Model for Drugs Administered Intravenously

More information

ACTIVITY: Deriving the Area Formula of a Trapezoid

ACTIVITY: Deriving the Area Formula of a Trapezoid 4.3 Areas of Trapezoids a trapezoid? How can you derive a formula for te area of ACTIVITY: Deriving te Area Formula of a Trapezoid Work wit a partner. Use a piece of centimeter grid paper. a. Draw any

More information

SAMPLE DESIGN FOR THE TERRORISM RISK INSURANCE PROGRAM SURVEY

SAMPLE DESIGN FOR THE TERRORISM RISK INSURANCE PROGRAM SURVEY ASA Section on Survey Researc Metods SAMPLE DESIG FOR TE TERRORISM RISK ISURACE PROGRAM SURVEY G. ussain Coudry, Westat; Mats yfjäll, Statisticon; and Marianne Winglee, Westat G. ussain Coudry, Westat,

More information

Notes: Most of the material in this chapter is taken from Young and Freedman, Chap. 12.

Notes: Most of the material in this chapter is taken from Young and Freedman, Chap. 12. Capter 6. Fluid Mecanics Notes: Most of te material in tis capter is taken from Young and Freedman, Cap. 12. 6.1 Fluid Statics Fluids, i.e., substances tat can flow, are te subjects of tis capter. But

More information

Computer Science and Engineering, UCSD October 7, 1999 Goldreic-Levin Teorem Autor: Bellare Te Goldreic-Levin Teorem 1 Te problem We æx a an integer n for te lengt of te strings involved. If a is an n-bit

More information

Theoretical calculation of the heat capacity

Theoretical calculation of the heat capacity eoretical calculation of te eat capacity Principle of equipartition of energy Heat capacity of ideal and real gases Heat capacity of solids: Dulong-Petit, Einstein, Debye models Heat capacity of metals

More information

Lecture 10. Limits (cont d) One-sided limits. (Relevant section from Stewart, Seventh Edition: Section 2.4, pp. 113.)

Lecture 10. Limits (cont d) One-sided limits. (Relevant section from Stewart, Seventh Edition: Section 2.4, pp. 113.) Lecture 10 Limits (cont d) One-sided its (Relevant section from Stewart, Sevent Edition: Section 2.4, pp. 113.) As you may recall from your earlier course in Calculus, we may define one-sided its, were

More information

Reinforced Concrete Beam

Reinforced Concrete Beam Mecanics of Materials Reinforced Concrete Beam Concrete Beam Concrete Beam We will examine a concrete eam in ending P P A concrete eam is wat we call a composite eam It is made of two materials: concrete

More information

SWITCH T F T F SELECT. (b) local schedule of two branches. (a) if-then-else construct A & B MUX. one iteration cycle

SWITCH T F T F SELECT. (b) local schedule of two branches. (a) if-then-else construct A & B MUX. one iteration cycle 768 IEEE RANSACIONS ON COMPUERS, VOL. 46, NO. 7, JULY 997 Compile-ime Sceduling of Dynamic Constructs in Dataæow Program Graps Soonoi Ha, Member, IEEE and Edward A. Lee, Fellow, IEEE Abstract Sceduling

More information

Writing Mathematics Papers

Writing Mathematics Papers Writing Matematics Papers Tis essay is intended to elp your senior conference paper. It is a somewat astily produced amalgam of advice I ave given to students in my PDCs (Mat 4 and Mat 9), so it s not

More information

Characterization of researchers in condensed matter physics using simple quantity based on h -index. M. A. Pustovoit

Characterization of researchers in condensed matter physics using simple quantity based on h -index. M. A. Pustovoit Caracterization of researcers in condensed matter pysics using simple quantity based on -index M. A. Pustovoit Petersburg Nuclear Pysics Institute, Gatcina Leningrad district 188300 Russia Analysis of

More information

SAT Subject Math Level 1 Facts & Formulas

SAT Subject Math Level 1 Facts & Formulas Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses

More information

TAKE-HOME EXP. # 2 A Calculation of the Circumference and Radius of the Earth

TAKE-HOME EXP. # 2 A Calculation of the Circumference and Radius of the Earth TAKE-HOME EXP. # 2 A Calculation of te Circumference and Radius of te Eart On two dates during te year, te geometric relationsip of Eart to te Sun produces "equinox", a word literally meaning, "equal nigt"

More information

2.1: The Derivative and the Tangent Line Problem

2.1: The Derivative and the Tangent Line Problem .1.1.1: Te Derivative and te Tangent Line Problem Wat is te deinition o a tangent line to a curve? To answer te diiculty in writing a clear deinition o a tangent line, we can deine it as te iting position

More information

Let's Learn About Notes

Let's Learn About Notes 2002 Product of Australia Contents Let's Learn About Notes by Beatrice Wilder Seet Seet 2 Seet 3 Seet 4 Seet 5 Seet 6 Seet 7 Seet 8 Seet 9 Seet 0 Seet Seet 2 Seet 3 Basic Information About Notes Lines

More information

In their classic textbook, Musgrave and Musgrave (1989)

In their classic textbook, Musgrave and Musgrave (1989) Integrating Expenditure and Tax Decisions Integrating Expenditure and Tax Decisions: Te Marginal Cost of Funds and te Marginal Benefit of Projects Abstract - Tis paper seeks to clarify te extent to wic

More information

2.13 Solid Waste Management. Introduction. Scope and Objectives. Conclusions

2.13 Solid Waste Management. Introduction. Scope and Objectives. Conclusions Introduction Te planning and delivery of waste management in Newfoundland and Labrador is te direct responsibility of municipalities and communities. Te Province olds overall responsibility for te development

More information

Comparison between two approaches to overload control in a Real Server: local or hybrid solutions?

Comparison between two approaches to overload control in a Real Server: local or hybrid solutions? Comparison between two approaces to overload control in a Real Server: local or ybrid solutions? S. Montagna and M. Pignolo Researc and Development Italtel S.p.A. Settimo Milanese, ITALY Abstract Tis wor

More information

Proposed experiment of which-way detection by longitudinal momentum transfer in Young's double slit experiment

Proposed experiment of which-way detection by longitudinal momentum transfer in Young's double slit experiment Proposed experiment of wic-way detection by longitudinal momentum transfer in Young's double slit experiment Masanori Sato Honda Electronics Co., Ltd., 20 Oyamazuka, Oiwa-co, Toyoasi, Aici 441-3193, Japan

More information

In other words the graph of the polynomial should pass through the points

In other words the graph of the polynomial should pass through the points Capter 3 Interpolation Interpolation is te problem of fitting a smoot curve troug a given set of points, generally as te grap of a function. It is useful at least in data analysis (interpolation is a form

More information

Warm medium, T H T T H T L. s Cold medium, T L

Warm medium, T H T T H T L. s Cold medium, T L Refrigeration Cycle Heat flows in direction of decreasing temperature, i.e., from ig-temperature to low temperature regions. Te transfer of eat from a low-temperature to ig-temperature requires a refrigerator

More information

Module 2. The Science of Surface and Ground Water. Version 2 CE IIT, Kharagpur

Module 2. The Science of Surface and Ground Water. Version 2 CE IIT, Kharagpur Module Te Science of Surface and Ground Water Version CE IIT, Karagpur Lesson 6 Principles of Ground Water Flow Version CE IIT, Karagpur Instructional Objectives On completion of te lesson, te student

More information

2.12 Student Transportation. Introduction

2.12 Student Transportation. Introduction Introduction Figure 1 At 31 Marc 2003, tere were approximately 84,000 students enrolled in scools in te Province of Newfoundland and Labrador, of wic an estimated 57,000 were transported by scool buses.

More information

The classical newsvendor model under normal demand with large coefficients of variation

The classical newsvendor model under normal demand with large coefficients of variation MPRA Munic Personal RePEc Arcive Te classical newsvendor model under normal demand wit large coefficients of variation George Halkos and Ilias Kevork University of Tessaly, Department of Economics August

More information

Computer Vision System for Tracking Players in Sports Games

Computer Vision System for Tracking Players in Sports Games Computer Vision System for Tracking Players in Sports Games Abstract Janez Perš, Stanislav Kovacic Faculty of Electrical Engineering, University of Lublana Tržaška 5, 000 Lublana anez.pers@kiss.uni-l.si,

More information

Strategic trading in a dynamic noisy market. Dimitri Vayanos

Strategic trading in a dynamic noisy market. Dimitri Vayanos LSE Researc Online Article (refereed) Strategic trading in a dynamic noisy market Dimitri Vayanos LSE as developed LSE Researc Online so tat users may access researc output of te Scool. Copyrigt and Moral

More information

Math Test Sections. The College Board: Expanding College Opportunity

Math Test Sections. The College Board: Expanding College Opportunity Taking te SAT I: Reasoning Test Mat Test Sections Te materials in tese files are intended for individual use by students getting ready to take an SAT Program test; permission for any oter use must be sougt

More information

A strong credit score can help you score a lower rate on a mortgage

A strong credit score can help you score a lower rate on a mortgage NET GAIN Scoring points for your financial future AS SEEN IN USA TODAY S MONEY SECTION, JULY 3, 2007 A strong credit score can elp you score a lower rate on a mortgage By Sandra Block Sales of existing

More information

2.23 Gambling Rehabilitation Services. Introduction

2.23 Gambling Rehabilitation Services. Introduction 2.23 Gambling Reabilitation Services Introduction Figure 1 Since 1995 provincial revenues from gambling activities ave increased over 56% from $69.2 million in 1995 to $108 million in 2004. Te majority

More information

Modeling User Perception of Interaction Opportunities for Effective Teamwork

Modeling User Perception of Interaction Opportunities for Effective Teamwork Modeling User Perception of Interaction Opportunities for Effective Teamwork Ece Kamar, Ya akov Gal and Barbara J. Grosz Scool of Engineering and Applied Sciences Harvard University, Cambridge, MA 02138

More information

Pre-trial Settlement with Imperfect Private Monitoring

Pre-trial Settlement with Imperfect Private Monitoring Pre-trial Settlement wit Imperfect Private Monitoring Mostafa Beskar University of New Hampsire Jee-Hyeong Park y Seoul National University July 2011 Incomplete, Do Not Circulate Abstract We model pretrial

More information

1 Derivatives of Piecewise Defined Functions

1 Derivatives of Piecewise Defined Functions MATH 1010E University Matematics Lecture Notes (week 4) Martin Li 1 Derivatives of Piecewise Define Functions For piecewise efine functions, we often ave to be very careful in computing te erivatives.

More information

4.1 Right-angled Triangles 2. 4.2 Trigonometric Functions 19. 4.3 Trigonometric Identities 36. 4.4 Applications of Trigonometry to Triangles 53

4.1 Right-angled Triangles 2. 4.2 Trigonometric Functions 19. 4.3 Trigonometric Identities 36. 4.4 Applications of Trigonometry to Triangles 53 ontents 4 Trigonometry 4.1 Rigt-angled Triangles 4. Trigonometric Functions 19 4.3 Trigonometric Identities 36 4.4 pplications of Trigonometry to Triangles 53 4.5 pplications of Trigonometry to Waves 65

More information

Analysis of Algorithms - Recurrences (cont.) -

Analysis of Algorithms - Recurrences (cont.) - Analysis of Algoritms - Recurrences (cont.) - Andreas Ermedal MRTC (Mälardalens Real Time Researc Center) andreas.ermedal@md.se Autumn 004 Recurrences: Reersal Recurrences appear frequently in running

More information

Cyber Epidemic Models with Dependences

Cyber Epidemic Models with Dependences Cyber Epidemic Models wit Dependences Maocao Xu 1, Gaofeng Da 2 and Souuai Xu 3 1 Department of Matematics, Illinois State University mxu2@ilstu.edu 2 Institute for Cyber Security, University of Texas

More information

Optimal and Autonomous Incentive-based Energy Consumption Scheduling Algorithm for Smart Grid

Optimal and Autonomous Incentive-based Energy Consumption Scheduling Algorithm for Smart Grid 1 Optimal and Autonomous Incentive-based Energy Consumption Sceduling Algoritm for Smart Grid Amir-Hamed Mosenian-Rad, Vincent W.S. Wong, Juri Jatskevic and Robert Scober Department of Electrical and Computer

More information

Chapter 10: Refrigeration Cycles

Chapter 10: Refrigeration Cycles Capter 10: efrigeration Cycles Te vapor compression refrigeration cycle is a common metod for transferring eat from a low temperature to a ig temperature. Te above figure sows te objectives of refrigerators

More information

A STUDY ON NUMERICAL COMPUTATION OF POTENTIAL DISTRIBUTION AROUND A GROUNDING ROD DRIVEN IN SOIL

A STUDY ON NUMERICAL COMPUTATION OF POTENTIAL DISTRIBUTION AROUND A GROUNDING ROD DRIVEN IN SOIL A STUDY ON NUMERICAL COMPUTATION OF POTENTIAL DISTRIBUTION AROUND A GROUNDING ROD DRIVEN IN SOIL Ersan Şentürk 1 Nurettin Umurkan Özcan Kalenderli 3 e-mail: ersan.senturk@turkcell.com.tr e-mail: umurkan@yildiz.edu.tr

More information

Guide to Cover Letters & Thank You Letters

Guide to Cover Letters & Thank You Letters Guide to Cover Letters & Tank You Letters 206 Strebel Student Center (315) 792-3087 Fax (315) 792-3370 TIPS FOR WRITING A PERFECT COVER LETTER Te resume never travels alone. Eac time you submit your resume

More information

Research on the Anti-perspective Correction Algorithm of QR Barcode

Research on the Anti-perspective Correction Algorithm of QR Barcode Researc on te Anti-perspective Correction Algoritm of QR Barcode Jianua Li, Yi-Wen Wang, YiJun Wang,Yi Cen, Guoceng Wang Key Laboratory of Electronic Tin Films and Integrated Devices University of Electronic

More information

College Planning Using Cash Value Life Insurance

College Planning Using Cash Value Life Insurance College Planning Using Cas Value Life Insurance CAUTION: Te advisor is urged to be extremely cautious of anoter college funding veicle wic provides a guaranteed return of premium immediately if funded

More information

Exam 2 Review. . You need to be able to interpret what you get to answer various questions.

Exam 2 Review. . You need to be able to interpret what you get to answer various questions. Exam Review Exam covers 1.6,.1-.3, 1.5, 4.1-4., and 5.1-5.3. You sould know ow to do all te omework problems from tese sections and you sould practice your understanding on several old exams in te exam

More information

Lab 3: Amplifier Circuits LAB 3 AMPLIFIER CIRCUITS. (Small-signal Low-frequency voltage amplifier) EC0222 Electronic Circuits Lab Manual

Lab 3: Amplifier Circuits LAB 3 AMPLIFIER CIRCUITS. (Small-signal Low-frequency voltage amplifier) EC0222 Electronic Circuits Lab Manual Lab 3: Amplifier ircuits 3. LAB 3 AMPLIFIR IRUITS xperiment 3.: xperiment 3.2: Transistor amplifier (Small-signal Low-frequency voltage amplifier) op-amp feedback amplifier 0222 lectronic ircuits Lab Manual

More information

Areas and Centroids. Nothing. Straight Horizontal line. Straight Sloping Line. Parabola. Cubic

Areas and Centroids. Nothing. Straight Horizontal line. Straight Sloping Line. Parabola. Cubic Constructing Sear and Moment Diagrams Areas and Centroids Curve Equation Sape Centroid (From Fat End of Figure) Area Noting Noting a x 0 Straigt Horizontal line /2 Straigt Sloping Line /3 /2 Paraola /4

More information