Learn to Recognize Sptil Structure Through Compre Recognition

Size: px
Start display at page:

Download "Learn to Recognize Sptil Structure Through Compre Recognition"

Transcription

1 Stroke-Bse Performnce Metrics for Hnwritten Mthemticl Expressions Richr Znii Amit Pilly eprtment of Computer Science Rochester Institute of Technology, NY, SA Hrol Mouchère Christin Vir-Guin LNAM, niversité e Nntes IRCCyN, Frnce orothe Blostein lostein@cs.queensu.c School of Computing, Queen s niversity, Cn Astrct Evluting mthemticl expression recognition involves complex interction of input primitives (e.g. pen/finger strokes), recognize symols, n recognize sptil structure. Existing performnce metrics simplify this prolem y seprting the ssessment of sptil structure from the ssessment of symol segmenttion n clssifiction. These metrics o not chrcterize the overll ccurcy of pense mthemtics recognition, mking it ifficult to compre mth recognition lgorithms, n preventing the use of mchine lerning lgorithms requiring criterion function chrcterizing overll system performnce. To ress this prolem, we introuce performnce metrics tht rige the gp from hnwritten strokes to sptil structure. Our metrics re compute using iprtite grphs tht represent clssifiction, segmenttion n sptil structure t the stroke level. Overll correctness of n expression is mesure y counting the numer of relelings of noes n eges neee to mke the iprtite grph for recognition result mtch the iprtite grph for groun truth. This metric my lso e use with other primitive types (e.g. imge pixels). Keywors-Performnce Evlution; Mth Recognition; Hnwriting Recognition; Grphics Recognition I. INTROCTION Evluting the performnce of ocument nlysis systems is n importnt n ifficult prolem. As recently summrize y Silv [1], much of the ifficulty stems from iversity in gols, input types, input omins, concepts, output grnulrity, evlution moments, n evlution metrics. Our work focuses on ressing performnce evlution issues tht rise ue to iversity in grnulrity. Performnce metrics re simpler to efine for computtions in which input n output items hve similr grnulrity. In the cse of mth recognition, the grnulrity iffers mrkely, with sptil rrngement of strokes or symols s input, n hierrchicl lyout escription (e.g. L A TEX, see Figure 1) n/or representtion of mening (e.g. Content MthML, OpenMth) s output. There is nee for stnr metrics tht permit meningful comprison of mth recognition results [2], [3], oth for compring systems, n for use with mchine lerning lgorithms tht optimize system performnce. Our primry contriution is iprtite grph-se representtion of expression structure t the level of primitives (e.g. strokes), tht cptures errors in oth recognize symols n lyout. We were motivte to use representtion t the primitive rther thn symol level, ecuse the istinction etween symol segmenttion n structure recognition is sometimes lurre. For exmple, the configurtion = cn e viewe either s single symol consisting of two sptilly seprte strokes, or s two symols (short horizontl lines) with one top nother. We consier isolte expressions in this pper, ut our metho my e pte in stright-forwr mnner for multiple expressions, flowchrts, tles, n even imges using pixel regions such s connecte components or imge ptches. The secon contriution of the pper is set of new error metrics se in our iprtite representtion. This inclues metrics tht chrcterize overll recognition ccurcy, proviing criterion function for expressions, se on strokes s primitives; existing criterion functions use symols s primitives, s iscusse in Section II. We ssume tht it is possile to efine groun truth interprettion for given set of strokes. This exclues unersegmente strokes, in which single stroke is use to prouce two items tht must e represente seprtely in \frc{}{ˆ} ) Input ) L A TEX c) Symol Lyout Tree (: up, : own, S: superscript) Figure 1. Hnwritten Expression Contining Five Strokes n Four Symols. A LATEX string () or equivlent Symol Lyout Tree (c) my e use to represent symol rrngement

2 the groun truth interprettion (e.g. for cursive x written using single stroke, ut contining two symols). For etile performnce nlysis, ifferent metrics re neee to chrcterize ccurcy for specific tsks: segmenttion, clssifiction, n prsing. However, in some situtions single vlue chrcterizing performnce is neee, such s when using mchine lerning lgorithms to optimize system performnce s whole. Thus, in this pper, we iscuss severl component metrics (section IV) n lso iscuss methos of comining these into single overll estimte of performnce (Section V). II. PREVIOS WORK IN EVALATING MATH RECOGNITION Mthemticl expression recognition is n ctive reserch fiel, oth for on-line n off-line t [4], [5], [6], Grounthruthe tset re now ville (e.g. [7] (off-line) n [8] (online)). As pointe out y oth Lpointe n Blostein [2] n Awl et l. [3], the mth recognition omin now nees stnr evlution metrics to support comprisons of existing n newly evelope systems. Most existing pproches to evluting mth recognition compute istnce etween the recognize expression n the groun truth, ccoring to ifferent spects. The expression recognition rte is common, ut glol n reltively uninformtive, s it counts only expressions tht precisely mtch groun truth. Symol recognition rte oes not consier symol lyout; Bseline recognition checks only if symols pper on the correct seline reltive to symol. Some metrics, such s the verge performnce inex [9] weight errors epening on the epth of nesting for selines in n expression. A ifficulty in efining n ccurcy metric for symol lyout in mth, is tht the tree-se representtion neee to represent symol lyout is unsuitle for use with clssicl metrics use for text recognition, such s the Levenstein eit istnce. One solution is to use tree-eit istnce, ut this is not use in prctice, ecuse of the NP complexity of the existing lgorithms to mtch oth tree eges n noes. An interesting solution y Grin et l. [10] proposes to trnsform the tree into token string which llows one to use eit istnce. The rwck of this pproch is tht it looses some of the eit opertions offere y tree eits (like swpping chilren of noe), leing them to incur high cost in the string-se representtion. Our min contriution is iprtite grph representtion of expression structure t the level of strokes, from which metrics se on Hmming istnces my e simply efine, with n intuitive interprettion. Given set of input primitives for test expression, our representtion prevents the nee to mtch iniviul strokes, so tht only stroke lels n lyout reltionships etween strokes nee to e mtche, s escrie in the next section. III. EXPRESSION REPRESENTATION In our pproch, the recognizer output n groun truth interprettions must first e converte into iprtite grphs. This is illustrte in Figures 2 n 3. The iprtite representtion is shown in Figure 3), where the noes of the grph represent ech stroke in the expression twice: s n unlele input stroke (t left), n with n ssigne symol lel n etecte reltionships (t right, with sptil reltionships shown s incoming eges). Note tht there re N(N 1) eges in this grph, where N is the numer of strokes, n we omit eges from strokes to themselves. For legiility, eges representing no reltionship re not rwn. This iprtite grph is constructe from symol lyout tree: strokes in symol noes re split into seprte stroke noes (see Figure 2), with ech stroke possessing the sptil reltionships of its ssocite symol. All stroke noes inherit the sptil reltionships of their ncestors in the lyout tree. In Figure 2, the two strokes corresponing to the inherit the own sptil reltionship of the singlestroke. Note tht this inheritnce pplies to ll sptil reltionships, incluing continment y squre roots, n horizontl jcency: for exmple, in k + m, m inherits the Right reltionship of the + reltive to the k. The informtion presente in the AG in Figure 2) cn then e converte irectly into iprtite grph, s shown in Figure 3). Note tht strokes with the sme set of incoming sptil reltionships in Figure 3 re symols, i.e. stroke reltionships inuce segmenttion of strokes into symols. It is esier to visulize ifferences in interprettions using the iprtite representtion (s noe positions in the grph my e fixe cross interprettions), ut it is esier to visulize interprete lyout from the AG representtion. To evlute symol segmenttion seprtely from lyout, we my use secon iprtite grph: see Figure 3). An unirecte ege is plce etween ll pirs of non-ienticl strokes elonging to symol. Thus symol compose of 3 strokes is represente y 6 eges; one isolte stroke corresponing to symol is not connecte. ) Stroke Lyout Tree ) Stroke Lyout AG Figure 2. Stroke-level Groun Truth Representtions for Expression in Figure 1). Strokes re lele using s<num>. In ), the stroke lyout tree is converte to AG y ing incoming sptil reltionships t noe to ll of its escenents. Sptil reltionships re represente y : up, : own, : superscript, Su: suscript, n R: right

3 ) Stroke lels n lyout ) Segmenttion Figure 3. Biprtite Grphs Representing the Expression in Figure 1. Noes represent strokes, n lele eges represent sptil reltionships. In ), symol clsses re shown using noe lels, n sptil reltionships using ege lels. This grph represents the sme informtion s in Figure 2). In ) segmenttion grph is shown, in which strokes elonging to symol re connecte IV. METRICS FOR SPECIFIC ERROR TYPES Given two iprtite grphs representing recognizer output n groun truth for strokes in hnwritten mth expression, recognition errors my e foun irectly s isgreeing noe or ege lels. A numer of exmples re provie in Figure 4; incorrect lels n reltionships reltive to groun truth re shown in re. ) mislele stroke (1 error, C ) ) misrecognize reltionships (2 errors, L ) c) segmenttion error, where the hs een split into two symols. There re two mislele strokes (clssifiction errors), n spurious sptil reltionship etween n (3 errors) ) error similr to tht in c), ut with the stem of the misrecognize s eing ove the frction line; there is lso missing reltionship etween n (5 errors) These errors re summrize in Tle I. Aitionlly, one my count the numer of isgreeing stroke pirings in segmenttion grphs s illustrte in Figure 3) ( S ). In Figure 4c) n ), two segmenttion grph eges from groun truth re missing. For set of strokes S n two expressions efine on S represente y iprtite grphs E 1 n E 2, we efine the metrics elow for specific stroke properties. Let e the set of ll non-ienticl stroke pirs: = {(p, q) S S p q}, where = S S 1. Ech metric elow is Hmming istnce, specificlly the numer of isgreeing lels/reltionships. As such, ech stisfies the four requirements for metric [11]: non-negtivity, symmetry ( (E 1, E 2 ) = (E 2, E 1 ), (E 1, E 1 ) = 0, n the tringle inequlity: (E 1, E 3 ) (E 1, E 2 ) + (E 2, E 3 ). Clssifiction ( C ): The numer of strokes with ifferent symol lels in the expression grphs E 1 n E 2 : C (E 1, E 2 ) = {s S l(s, E 1 ) l(s, E 2 )} (1) Lyout ( L ): Let L 1 n L 2 e the set of lelle eges in expression grphs E 1 n E 2. Lyout isgreement is the numer of isgreeing ege lels etween non-ienticl strokes: L (E 1, E 2 ) = L 1 L 2 (2) Segmenttion ( S ): This is efine similrly to lyout, ut using unirecte segmenttion iprtite grphs (see Figure 3) B 1 n B 2 constructe on the set of strokes for ech symol reltion tree. S (E 1, E 2 ) = B 1 B 2 (3) V. EXPRESSION-LEVEL ISTANCE METRICS We now comine our metrics for specific error types (clssifiction, segmenttion, n lyout) into Expression- Level istnce Metrics tht efine single istnce mesure for two interprettions of n expression. First consier the istnce metric B [0, 1] efine s the numer of isgreeing stroke lels n sptil reltionships, such s shown in the thumnil imges of Figure 4. This is Hmming istnce, with S 2 elements in ech vector of noe/ege lels for grph. B (E 1, E 2 ) = C + L S 2 (4) This metric is unweighte, n s result will prouce less istnce for clssifiction errors thn errors in lyout n segmenttion (represente implicitly in the lyout reltionships). As n solute mesure of the ifference etween two iprtite grphs B is sufficient, ut one my wnt to weight errors to mke clssifiction errors proportionl to segmenttion n lyout errors. In prticulr, when compring lgorithms for use in prctice, or when using mchine lerning to optimize the complete recognition system, one my wnt to weight the ifferent error types. We efine metric E [0, 1] s the verge per-stroke clssifiction, segmenttion n lyout errors: C (E 1,E 2 ) S + S (E 1,E 2 ) + L (E 1,E 2 ) E(E 1, E 2) = 3 (5) We use the squre root of the segmenttion n sptil reltionship istnces in orer to mke them proportionl to S rther thn S 2 (one coul inste ivie L n S

4 6 6 R R ) Clssifiction Error ( 6) ) Lyout Error ( R) 0 R Su 1 0 c) Clssifiction n Segmenttion ( {0,1}) ) Clssifiction, Segmenttion n Lyout Figure 4. Exmple Recognition Errors for Expression in Figure 1. In the AGs errors re shown in re, n y fille noes n eges in the iprtite grphs. In the iprtite grph thumnils, noes correspon to strokes s shown in Figure 3. In prt ) there re five errors: the hs een seprte into two mis-clssifie strokes, with two spurious sptil reltionships ( n Su), n one missing reltionship (the superscript etween the n the verticl line in the ). Tle I ISTANCE BETWEEN EXPRESSIONS IN FIGRE 4 AN GRON-TRTH IN FIGRES 2B) AN 3A) Fig. 4 C S L B E ) ) c) ) y S 1 for the sme reson). This prevents ifferences in segments n sptil reltionships from eing weighte less hevily thn ifferences in stroke (symol) clssifiction lels. As ech component istnce is in [0, 1], E lso lies in the intervl [0, 1]. B n E re proper metrics. They re non-negtive, symmetric, n the istnce from lyout tree to itself is 0. As the squre root of non-negtive vlues is n orerpreserving monotonic function, the squre root of metric is lso metric. Given tht C, S n L re proper metrics, their sum oeys the tringle inequlity y efinition. Similrly, using the verge of their sum oes not invlite the metric property. Both B n E require O( S 2 ) time to compute. In prctice, S tens to e reltively smll, n so the qurtic complexity is not significnt concern. Further, sent sptil reltionships nee never e explicitly compre: we cn simply count lels n reltionships present in t lest one of the two input grphs. In orer to illustrte n compre these metrics, Tle I shows these five istnces etween errors n its grountruth. Notice tht clssifiction errors re weighte more hevily, n tht in generl the compute istnce/error vlue is higher for E thn B. VI. GENERALIZATION: SEGMENTS AN PIXELS We hve ssume tht no stroke correspons to more thn one symol in the input (i.e. no stroke is uner-segmente). This ssumption my e remove if we use finer-grine primitives, such s line segments rther thn whole strokes. A single stroke contining the two symols x cn then e prtitione, n the resulting segmenttion evlute. ocument imges often hve some symols overlpping within single connecte component, such s in: y x, where the frction line n y my intersect. In this cse we cn econstruct connecte components into smller sucomponents tht correspon to smll contiguous regions, or s more extreme pproch, tking pixels to e the primitives. sing the smllest possile primitives (e.g. pixels) is ttrctive ecuse uner-segmenttion cnnot occur; however, efficiency my ecome prolem, s the iprtite grphs/ags woul e very lrge. Pixel-level groun truth is imprecise; however, this level of groun truthing is common in computer vision, where it is unerstoo tht the humn interprettion involve in constructing groun truth results in resiul errors for ecisions within miguous regions (e.g.

5 ientifying the specific split point etween two connecte symols rwn with single stroke (e.g. x)). With pproprite primitives, the metrics presente my e use s criterion functions for mchine lerning lgorithms. In most cses, losses for errors in stroke leling n reltionships will nee to e softene to vlues in [0, 1] rther thn {0, 1}, e.g. to voi iscontinuities in the error surfce when using lgorithms se on grient escent. These soft errors my e otine using itionl metrics for stroke lels n reltionships (e.g. proilities or fuzzy vlues). VII. CONCLSION We hve presente new metrics for compring the similrity of two interprettions of set of online strokes, with ppliction to pen-se mthemtics recognition. Our pproch is novel in tht it uses strokes rther thn symols s the sis for compring symol n structure recognition results. This hs the vntge of proviing roer chrcteriztion of system performnce, llowing expressionlevel performnce to e ssesse in terms of input primitives. Our metrics cn e efficiently compute, in time O(n 2 ), where n is the numer of strokes. Note tht hnwritten expressions typiclly consist of reltively smll numer of strokes. The pproch cn lso e esily pte other pen-se omins, such s recognition of flowchrts, n for use in imges. An open question is whether the metric cn e usefully pplie when one cnnot ssume tht the sets primitives for two recognition results eing compre mtch (e.g. for Mthemticl Informtion Retrievl (MIR) pplictions). A relte issue is efining metrics for evlution of mthemticl content (i.e. mthemticl syntx of recognize expression); the metho presente in this pper resses only evlution of lyout. As mthemticl content is normlly represente hierrchiclly y opertor trees, it my e possile to employ iprtite grph-se pproch to evlution there s well, gin using input primitives s the noes in the grph. [3] A.-M. Awl, H. Mouchere, n C. Vir-Guin, The prolem of hnwritten mthemticl expression recognition evlution, in Int l Conf. on Frontiers in Hnwriting Recognition, Kolkt, Ini, 2010, pp [4]. Blostein n A. Grvec, Recognition of mthemticl nottion, in Hnook of Chrcter Recognition n ocument Imge Anlysis. Worl Scientific Pulishing Compny, 1997, pp [5] K.-F. Chn n.-y. Yeung, Mthemticl expression recognition: survey, Interntionl Journl on ocument Anlysis n Recognition, vol. 3, pp. 3 15, Aug [6]. Grin n B. Chuhuri, OCR of Printe Mthemticl Expressions. Springer, 2007, pp [7] S. chi, A. Nomur, n M. Suzuki, Quntittive nlysis of mthemticl ocuments, Int l J. ocument Anlysis n Recognition, vol. 7, no. 4, pp , [8] S. McLen, G. Lhn, E. Lnk, M. Mrzouk, n. Tusky, Grmmr-se techniques for creting grountruthe sketch corpor, Int l. J. ocument Anlysis n Recognition, vol. 14, no. 1, pp , [9]. Grin n B. Chuhuri, A corpus for OCR reserch on mthemticl expressions, Int l J. ocument Anlysis n Recognition, vol. 7, no. 4, pp , [10] K. Sin, A. sgupt, n. Grin, Emers: tree mtching-se performnce evlution of mthemticl expression recognition systems, Int l J. ocument Anlysis n Recognition, no. 14, pp , [11] R. u, P. Hrt, n. Stork, Pttern Clssifiction, 2n e. Wiley, Acknowlegements: This mteril is se upon work supporte y the Ntionl Science Fountion uner Grnt No. IIS , the Nturl Sciences n Engineering Reserch Council of Cn, the Xerox Fountion, n the Center for Emerging n Innovtive Sciences (NYSTAR). REFERENCES [1] A. Sliv, Metrics for evluting performnce in ocument nlysis: ppliction to tles, Int l J. ocument Anlysis n Recognition, vol. 14, pp , [2] A. Lpointe n. Blostein, Issues in performnce evlution: A cse stuy of mth recognition. IEEE Computer Society, 2009, pp

Reasoning to Solve Equations and Inequalities

Reasoning to Solve Equations and Inequalities Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing

More information

1.2 The Integers and Rational Numbers

1.2 The Integers and Rational Numbers .2. THE INTEGERS AND RATIONAL NUMBERS.2 The Integers n Rtionl Numers The elements of the set of integers: consist of three types of numers: Z {..., 5, 4, 3, 2,, 0,, 2, 3, 4, 5,...} I. The (positive) nturl

More information

Section 5-4 Trigonometric Functions

Section 5-4 Trigonometric Functions 5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form

More information

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered: Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you

More information

EQUATIONS OF LINES AND PLANES

EQUATIONS OF LINES AND PLANES EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint

More information

Rotating DC Motors Part II

Rotating DC Motors Part II Rotting Motors rt II II.1 Motor Equivlent Circuit The next step in our consiertion of motors is to evelop n equivlent circuit which cn be use to better unerstn motor opertion. The rmtures in rel motors

More information

Regular Sets and Expressions

Regular Sets and Expressions Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite

More information

Homework 3 Solutions

Homework 3 Solutions CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

More information

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( ) Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

More information

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324 A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................

More information

Or more simply put, when adding or subtracting quantities, their uncertainties add.

Or more simply put, when adding or subtracting quantities, their uncertainties add. Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re

More information

Pentominoes. Pentominoes. Bruce Baguley Cascade Math Systems, LLC. The pentominoes are a simple-looking set of objects through which some powerful

Pentominoes. Pentominoes. Bruce Baguley Cascade Math Systems, LLC. The pentominoes are a simple-looking set of objects through which some powerful Pentominoes Bruce Bguley Cscde Mth Systems, LLC Astrct. Pentominoes nd their reltives the polyominoes, polycues, nd polyhypercues will e used to explore nd pply vrious importnt mthemticl concepts. In this

More information

An Undergraduate Curriculum Evaluation with the Analytic Hierarchy Process

An Undergraduate Curriculum Evaluation with the Analytic Hierarchy Process An Undergrdute Curriculum Evlution with the Anlytic Hierrchy Process Les Frir Jessic O. Mtson Jck E. Mtson Deprtment of Industril Engineering P.O. Box 870288 University of Albm Tuscloos, AL. 35487 Abstrct

More information

1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator

1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.

More information

4.11 Inner Product Spaces

4.11 Inner Product Spaces 314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces

More information

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers. 2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this

More information

Babylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity

Babylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity Bbylonin Method of Computing the Squre Root: Justifictions Bsed on Fuzzy Techniques nd on Computtionl Complexity Olg Koshelev Deprtment of Mthemtics Eduction University of Texs t El Pso 500 W. University

More information

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions. Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd

More information

Bypassing Space Explosion in Regular Expression Matching for Network Intrusion Detection and Prevention Systems

Bypassing Space Explosion in Regular Expression Matching for Network Intrusion Detection and Prevention Systems Bypssing Spce Explosion in Regulr Expression Mtching for Network Intrusion Detection n Prevention Systems Jignesh Ptel, Alex Liu n Eric Torng Dept. of Computer Science n Engineering Michign Stte University

More information

Evaluation of Prediction Models for Marketing Campaigns

Evaluation of Prediction Models for Marketing Campaigns Evlution of Preiction Moels for Mrketing Cmpigns Shron Rosset Amocs Lt n Stnfor University shronr@mocscom Nurit Vtnik Amocs (Isrel Lt nuritv@mocscom Eint Neumnn Amocs (Isrel Lt n Tel-Aviv University eintn@mocscom

More information

Experiment 6: Friction

Experiment 6: Friction Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology

More information

0.1 Basic Set Theory and Interval Notation

0.1 Basic Set Theory and Interval Notation 0.1 Bsic Set Theory nd Intervl Nottion 3 0.1 Bsic Set Theory nd Intervl Nottion 0.1.1 Some Bsic Set Theory Notions Like ll good Mth ooks, we egin with definition. Definition 0.1. A set is well-defined

More information

6.2 Volumes of Revolution: The Disk Method

6.2 Volumes of Revolution: The Disk Method mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine so-clled volumes of

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology

More information

Basic Analysis of Autarky and Free Trade Models

Basic Analysis of Autarky and Free Trade Models Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently

More information

How To Network A Smll Business

How To Network A Smll Business Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology

More information

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one. 5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued

More information

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn 33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of

More information

The Principle of No Punishment Without a Law for It LEARNING OBJECTIVES: CRLA.GAAN:15.01.07-01.07

The Principle of No Punishment Without a Law for It LEARNING OBJECTIVES: CRLA.GAAN:15.01.07-01.07 True / Flse 1. An ex post fcto lw is lw which hs retroctive effect.. True b. Flse True 2. An lcoholic cnnot be convicte for the offense of being runk in public plce bse upon the Eighth n Fourteenth Amenments..

More information

Binary Representation of Numbers Autar Kaw

Binary Representation of Numbers Autar Kaw Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology

More information

DATABASDESIGN FÖR INGENJÖRER - 1056F

DATABASDESIGN FÖR INGENJÖRER - 1056F DATABASDESIGN FÖR INGENJÖRER - 06F Sommr 00 En introuktionskurs i tssystem http://user.it.uu.se/~ul/t-sommr0/ lt. http://www.it.uu.se/eu/course/homepge/esign/st0/ Kjell Orsorn (Rusln Fomkin) Uppsl Dtse

More information

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the

More information

Unit 6: Exponents and Radicals

Unit 6: Exponents and Radicals Eponents nd Rdicls -: The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N): - counting numers. {,,,,, } Whole Numers (W): - counting numers with 0. {0,,,,,, } Integers (I): -

More information

CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001

CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001 CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic

More information

Integration by Substitution

Integration by Substitution Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is

More information

Angles 2.1. Exercise 2.1... Find the size of the lettered angles. Give reasons for your answers. a) b) c) Example

Angles 2.1. Exercise 2.1... Find the size of the lettered angles. Give reasons for your answers. a) b) c) Example 2.1 Angles Reognise lternte n orresponing ngles Key wors prllel lternte orresponing vertilly opposite Rememer, prllel lines re stright lines whih never meet or ross. The rrows show tht the lines re prllel

More information

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive

More information

All pay auctions with certain and uncertain prizes a comment

All pay auctions with certain and uncertain prizes a comment CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 1-2015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin

More information

Scalable Mining of Large Disk-based Graph Databases

Scalable Mining of Large Disk-based Graph Databases Sclle Mining of Lrge Disk-sed Grph Dtses Chen Wng Wei Wng Jin Pei Yongti Zhu Bile Shi Fudn University, Chin, {chenwng, weiwng1, 2465, shi}@fudn.edu.cn Stte University of New York t Bufflo, USA & Simon

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology

More information

Graphs on Logarithmic and Semilogarithmic Paper

Graphs on Logarithmic and Semilogarithmic Paper 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

More information

Helicopter Theme and Variations

Helicopter Theme and Variations Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the

More information

The art of Paperarchitecture (PA). MANUAL

The art of Paperarchitecture (PA). MANUAL The rt of Pperrhiteture (PA). MANUAL Introution Pperrhiteture (PA) is the rt of reting three-imensionl (3D) ojets out of plin piee of pper or ror. At first, esign is rwn (mnully or printe (using grphil

More information

The remaining two sides of the right triangle are called the legs of the right triangle.

The remaining two sides of the right triangle are called the legs of the right triangle. 10 MODULE 6. RADICAL EXPRESSIONS 6 Pythgoren Theorem The Pythgoren Theorem An ngle tht mesures 90 degrees is lled right ngle. If one of the ngles of tringle is right ngle, then the tringle is lled right

More information

Small Businesses Decisions to Offer Health Insurance to Employees

Small Businesses Decisions to Offer Health Insurance to Employees Smll Businesses Decisions to Offer Helth Insurnce to Employees Ctherine McLughlin nd Adm Swinurn, June 2014 Employer-sponsored helth insurnce (ESI) is the dominnt source of coverge for nonelderly dults

More information

Modular Generic Verification of LTL Properties for Aspects

Modular Generic Verification of LTL Properties for Aspects Modulr Generic Verifiction of LTL Properties for Aspects Mx Goldmn Shmuel Ktz Computer Science Deprtment Technion Isrel Institute of Technology {mgoldmn, ktz}@cs.technion.c.il ABSTRACT Aspects re seprte

More information

2 DIODE CLIPPING and CLAMPING CIRCUITS

2 DIODE CLIPPING and CLAMPING CIRCUITS 2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of

More information

MATH PLACEMENT REVIEW GUIDE

MATH PLACEMENT REVIEW GUIDE MATH PLACEMENT REVIEW GUIDE This guie is intene s fous for your review efore tking the plement test. The questions presente here my not e on the plement test. Although si skills lultor is provie for your

More information

** Dpt. Chemical Engineering, Kasetsart University, Bangkok 10900, Thailand

** Dpt. Chemical Engineering, Kasetsart University, Bangkok 10900, Thailand Modelling nd Simultion of hemicl Processes in Multi Pulse TP Experiment P. Phnwdee* S.O. Shekhtmn +. Jrungmnorom** J.T. Gleves ++ * Dpt. hemicl Engineering, Ksetsrt University, Bngkok 10900, Thilnd + Dpt.hemicl

More information

DlNBVRGH + Sickness Absence Monitoring Report. Executive of the Council. Purpose of report

DlNBVRGH + Sickness Absence Monitoring Report. Executive of the Council. Purpose of report DlNBVRGH + + THE CITY OF EDINBURGH COUNCIL Sickness Absence Monitoring Report Executive of the Council 8fh My 4 I.I...3 Purpose of report This report quntifies the mount of working time lost s result of

More information

How To Set Up A Network For Your Business

How To Set Up A Network For Your Business Why Network is n Essentil Productivity Tool for Any Smll Business TechAdvisory.org SME Reports sponsored by Effective technology is essentil for smll businesses looking to increse their productivity. Computer

More information

Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.

Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3. The nlysis of vrince (ANOVA) Although the t-test is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the t-test cn be used to compre the mens of only

More information

Gaze Manipulation for One-to-one Teleconferencing

Gaze Manipulation for One-to-one Teleconferencing Gze Mnipultion for One-to-one Teleconferencing A. Criminisi, J. Shotton, A. Blke, P.H.S. Torr Microsoft Reserch Ltd, Cmridge, UK Astrct A new lgorithm is proposed for novel view genertion in one-toone

More information

Automated Grading of DFA Constructions

Automated Grading of DFA Constructions Automted Grding of DFA Constructions Rjeev Alur nd Loris D Antoni Sumit Gulwni Dileep Kini nd Mhesh Viswnthn Deprtment of Computer Science Microsoft Reserch Deprtment of Computer Science University of

More information

Gene Expression Programming: A New Adaptive Algorithm for Solving Problems

Gene Expression Programming: A New Adaptive Algorithm for Solving Problems Gene Expression Progrmming: A New Adptive Algorithm for Solving Prolems Cândid Ferreir Deprtmento de Ciêncis Agráris Universidde dos Açores 9701-851 Terr-Chã Angr do Heroísmo, Portugl Complex Systems,

More information

The Velocity Factor of an Insulated Two-Wire Transmission Line

The Velocity Factor of an Insulated Two-Wire Transmission Line The Velocity Fctor of n Insulted Two-Wire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the

More information

#A12 INTEGERS 13 (2013) THE DISTRIBUTION OF SOLUTIONS TO XY = N (MOD A) WITH AN APPLICATION TO FACTORING INTEGERS

#A12 INTEGERS 13 (2013) THE DISTRIBUTION OF SOLUTIONS TO XY = N (MOD A) WITH AN APPLICATION TO FACTORING INTEGERS #A1 INTEGERS 13 (013) THE DISTRIBUTION OF SOLUTIONS TO XY = N (MOD A) WITH AN APPLICATION TO FACTORING INTEGERS Michel O. Rubinstein 1 Pure Mthemtics, University of Wterloo, Wterloo, Ontrio, Cnd mrubinst@uwterloo.c

More information

How To Understand The Theory Of Inequlities

How To Understand The Theory Of Inequlities Ostrowski Type Inequlities nd Applictions in Numericl Integrtion Edited By: Sever S Drgomir nd Themistocles M Rssis SS Drgomir) School nd Communictions nd Informtics, Victori University of Technology,

More information

Concept Formation Using Graph Grammars

Concept Formation Using Graph Grammars Concept Formtion Using Grph Grmmrs Istvn Jonyer, Lwrence B. Holder nd Dine J. Cook Deprtment of Computer Science nd Engineering University of Texs t Arlington Box 19015 (416 Ytes St.), Arlington, TX 76019-0015

More information

Operations with Polynomials

Operations with Polynomials 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply

More information

Capacitance and Dielectrics

Capacitance and Dielectrics 2.2 This is the Nerest One He 803 P U Z Z L E R Mny electronic components crry wrning lel like this one. Wht is there insie these evices tht mkes them so ngerous? Why wouln t you e sfe if you unplugge

More information

Value Function Approximation using Multiple Aggregation for Multiattribute Resource Management

Value Function Approximation using Multiple Aggregation for Multiattribute Resource Management Journl of Mchine Lerning Reserch 9 (2008) 2079-2 Submitted 8/08; Published 0/08 Vlue Function Approximtion using Multiple Aggregtion for Multittribute Resource Mngement Abrhm George Wrren B. Powell Deprtment

More information

Distributions. (corresponding to the cumulative distribution function for the discrete case).

Distributions. (corresponding to the cumulative distribution function for the discrete case). Distributions Recll tht n integrble function f : R [,] such tht R f()d = is clled probbility density function (pdf). The distribution function for the pdf is given by F() = (corresponding to the cumultive

More information

DAGmaps: Space Filling Visualization of Directed Acyclic Graphs

DAGmaps: Space Filling Visualization of Directed Acyclic Graphs Journl of Grph Algorithms nd Applictions http://jg.info/ vol. 13, no. 3, pp. 319 347 (2009) DAGmps: Spce Filling Visuliztion of Directed Acyclic Grphs Vssilis Tsirs 1,2 Sofi Trintfilou 1,2 Ionnis G. Tollis

More information

Gaze Manipulation for One-to-one Teleconferencing

Gaze Manipulation for One-to-one Teleconferencing Gze Mnipultion for One-to-one Teleconferencing A. Criminisi, J. Shotton, A. Blke, P.H.S. Torr Microsoft Reserch Ltd, Cmridge, UK Astrct A new lgorithm is proposed for novel view genertion in one-toone

More information

Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right.

Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right. Order of Opertions r of Opertions Alger P lese Prenthesis - Do ll grouped opertions first. E cuse Eponents - Second M D er Multipliction nd Division - Left to Right. A unt S hniqu Addition nd Sutrction

More information

CHAPTER 11 Numerical Differentiation and Integration

CHAPTER 11 Numerical Differentiation and Integration CHAPTER 11 Numericl Differentition nd Integrtion Differentition nd integrtion re bsic mthemticl opertions with wide rnge of pplictions in mny res of science. It is therefore importnt to hve good methods

More information

Radius of the Earth - Radii Used in Geodesy James R. Clynch February 2006

Radius of the Earth - Radii Used in Geodesy James R. Clynch February 2006 dius of the Erth - dii Used in Geodesy Jmes. Clynch Februry 006 I. Erth dii Uses There is only one rdius of sphere. The erth is pproximtely sphere nd therefore, for some cses, this pproximtion is dequte.

More information

1 Fractions from an advanced point of view

1 Fractions from an advanced point of view 1 Frtions from n vne point of view We re going to stuy frtions from the viewpoint of moern lger, or strt lger. Our gol is to evelop eeper unerstning of wht n men. One onsequene of our eeper unerstning

More information

Rotational Equilibrium: A Question of Balance

Rotational Equilibrium: A Question of Balance Prt of the IEEE Techer In-Service Progrm - Lesson Focus Demonstrte the concept of rottionl equilirium. Lesson Synopsis The Rottionl Equilirium ctivity encourges students to explore the sic concepts of

More information

A Conditional Model of Deduplication for Multi-Type Relational Data

A Conditional Model of Deduplication for Multi-Type Relational Data A Conditionl Model of Dedupliction for Multi-Type Reltionl Dt Aron Culott, Andrew McCllum Deprtment of Computer Science University of Msschusetts Amherst, MA 01003 {culott, mccllum}@cs.umss.edu Astrct

More information

LOW-COST, HIGH-IMPACT VIDEO PRODUCTION TECHNIQUES FOR LABORATORY INSTRUCTIONAL MATERIALS

LOW-COST, HIGH-IMPACT VIDEO PRODUCTION TECHNIQUES FOR LABORATORY INSTRUCTIONAL MATERIALS LOW-COST, HIGH-IMPCT VIEO PROUCTION TECHNIQUES FOR LBORTORY INSTRUCTIONL MTERILS Ewr R. oering 1 bstrct Vieo prouction techniques hve been evelope to prouce lbortory instructionl mterils tht re low cost,

More information

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors

More information

FAULT TREES AND RELIABILITY BLOCK DIAGRAMS. Harry G. Kwatny. Department of Mechanical Engineering & Mechanics Drexel University

FAULT TREES AND RELIABILITY BLOCK DIAGRAMS. Harry G. Kwatny. Department of Mechanical Engineering & Mechanics Drexel University SYSTEM FAULT AND Hrry G. Kwtny Deprtment of Mechnicl Engineering & Mechnics Drexel University OUTLINE SYSTEM RBD Definition RBDs nd Fult Trees System Structure Structure Functions Pths nd Cutsets Reliility

More information

Numeracy across the Curriculum in Key Stages 3 and 4. Helpful advice and suggested resources from the Leicestershire Secondary Mathematics Team

Numeracy across the Curriculum in Key Stages 3 and 4. Helpful advice and suggested resources from the Leicestershire Secondary Mathematics Team Numercy cross the Curriculum in Key Stges 3 nd 4 Helpful dvice nd suggested resources from the Leicestershire Secondry Mthemtics Tem 1 Contents pge The development of whole school policy 3 A definition

More information

Exam 1 Study Guide. Differentiation and Anti-differentiation Rules from Calculus I

Exam 1 Study Guide. Differentiation and Anti-differentiation Rules from Calculus I Exm Stuy Guie Mth 2020 - Clculus II, Winter 204 The following is list of importnt concepts from ech section tht will be teste on exm. This is not complete list of the mteril tht you shoul know for the

More information

Note: Principal version Modification Amendment Equivalence list Consolidated version from October 1 2014

Note: Principal version Modification Amendment Equivalence list Consolidated version from October 1 2014 Note: The following curriculum is consolidted version. It is leglly non-inding nd for informtionl purposes only. The leglly inding versions re found in the University of Innsruck Bulletins (in Germn).

More information

Decision Rule Extraction from Trained Neural Networks Using Rough Sets

Decision Rule Extraction from Trained Neural Networks Using Rough Sets Decision Rule Extrction from Trined Neurl Networks Using Rough Sets Alin Lzr nd Ishwr K. Sethi Vision nd Neurl Networks Lbortory Deprtment of Computer Science Wyne Stte University Detroit, MI 48 ABSTRACT

More information

Integration. 148 Chapter 7 Integration

Integration. 148 Chapter 7 Integration 48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but

More information

Hillsborough Township Public Schools Mathematics Department Computer Programming 1

Hillsborough Township Public Schools Mathematics Department Computer Programming 1 Essentil Unit 1 Introduction to Progrmming Pcing: 15 dys Common Unit Test Wht re the ethicl implictions for ming in tody s world? There re ethicl responsibilities to consider when writing computer s. Citizenship,

More information

Source Code verification Using Logiscope and CodeReducer. Christophe Peron Principal Consultant Kalimetrix

Source Code verification Using Logiscope and CodeReducer. Christophe Peron Principal Consultant Kalimetrix Source Code verifiction Using Logiscope nd CodeReducer Christophe Peron Principl Consultnt Klimetrix Agend Introducing Logiscope: Improving confidence nd developer s productivity Bsed on stte-of-the-rt

More information

Section 7-4 Translation of Axes

Section 7-4 Translation of Axes 62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the

More information

Online Multicommodity Routing with Time Windows

Online Multicommodity Routing with Time Windows Konrd-Zuse-Zentrum für Informtionstechnik Berlin Tkustrße 7 D-14195 Berlin-Dhlem Germny TOBIAS HARKS 1 STEFAN HEINZ MARC E. PFETSCH TJARK VREDEVELD 2 Online Multicommodity Routing with Time Windows 1 Institute

More information

COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE. Skandza, Stockholm ABSTRACT

COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE. Skandza, Stockholm ABSTRACT COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE Skndz, Stockholm ABSTRACT Three methods for fitting multiplictive models to observed, cross-clssified

More information

Solving BAMO Problems

Solving BAMO Problems Solving BAMO Problems Tom Dvis tomrdvis@erthlink.net http://www.geometer.org/mthcircles Februry 20, 2000 Abstrct Strtegies for solving problems in the BAMO contest (the By Are Mthemticl Olympid). Only

More information

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of

More information

Revisions published in the University of Innsbruck Bulletin of 18 June 2014, Issue 31, No. 509

Revisions published in the University of Innsbruck Bulletin of 18 June 2014, Issue 31, No. 509 Plese note: The following curriculum is for informtion purposes only nd not leglly inding. The leglly inding version is pulished in the pertinent University of Innsruck Bulletins. Originl version pulished

More information

Economics Letters 65 (1999) 9 15. macroeconomists. a b, Ruth A. Judson, Ann L. Owen. Received 11 December 1998; accepted 12 May 1999

Economics Letters 65 (1999) 9 15. macroeconomists. a b, Ruth A. Judson, Ann L. Owen. Received 11 December 1998; accepted 12 May 1999 Economics Letters 65 (1999) 9 15 Estimting dynmic pnel dt models: guide for q mcroeconomists b, * Ruth A. Judson, Ann L. Owen Federl Reserve Bord of Governors, 0th & C Sts., N.W. Wshington, D.C. 0551,

More information

Algebra Review. How well do you remember your algebra?

Algebra Review. How well do you remember your algebra? Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then

More information

MATH 150 HOMEWORK 4 SOLUTIONS

MATH 150 HOMEWORK 4 SOLUTIONS MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive

More information

SPECIAL PRODUCTS AND FACTORIZATION

SPECIAL PRODUCTS AND FACTORIZATION MODULE - Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come

More information

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding 1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde

More information

. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2

. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2 7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6

More information

Morgan Stanley Ad Hoc Reporting Guide

Morgan Stanley Ad Hoc Reporting Guide spphire user guide Ferury 2015 Morgn Stnley Ad Hoc Reporting Guide An Overview For Spphire Users 1 Introduction The Ad Hoc Reporting tool is ville for your reporting needs outside of the Spphire stndrd

More information

Technical Appendix: Multi-Product Firms and Trade Liberalization (Not For Publication)

Technical Appendix: Multi-Product Firms and Trade Liberalization (Not For Publication) Technicl Appenix: Multi-Prouct Firms n Tre Liberlition (Not For Publiction) Anrew B. Bernr Tuck School of Business t Drtmouth, CEPR & NBER Stephen J. Reing Princeton University & CEPR Peter K. Schott Yle

More information

Physics 43 Homework Set 9 Chapter 40 Key

Physics 43 Homework Set 9 Chapter 40 Key Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x

More information

How To Write A Recipe Card

How To Write A Recipe Card Interntionl Journl of Forecsting 18 (2002) 625 646 www.elsevier.com/ locte/ ijforecst A genetic lgorithms pproch to growth phse forecsting of wireless suscriers 1,2 Rjkumr Venktesn, V. Kumr* ING Center

More information