Motivation (Cont d) Motivation (Cont d)
|
|
- Maud Taylor
- 6 years ago
- Views:
Transcription
1 Data-Flow Analyi or Hot-Spot Program Optimization Jn Knoop Tchnich Univrität Win 1 Motivation 2 A Poibly Hug Program Rgion x Hot Spot 1 Hot Spot 2 A Poibly Hug Program Rgion 3 4 A Poibly Hug Program Rgion x x A Poibly Hug Program Rgion 5 6 Availability at a Singl Program Point Why Standard Data-Flow Analyi Fail a) b) Availability o trm al i n = AVAIL(n) = [[ (m,n) ]](AVAIL(n)) othrwi m prd(n) a := b := a := b := whr [[ (m,n) ]](b) = (COMP (m,n) + b) TRANSP (m,n) b := a+b b := a+b 7 8
2 Outlin o th Talk Standard v. Rvr Data-Flow Analyi Background Ential Th Conncting Link Th Clou: Why do it work? Application Concluion 9 Background Dmand-Drivn Data-Flow Analyi Agrawal (2000) Horwitz, Rp, Sagiv (1994+) Dutrwald, Gupta, Soa (1995+) Knoop (Euro-Par 1999, KPS 2007) 10 Rvr Data-Flow Analyi: Th Baic (Standard) Data-Flow Analyi Data-Flow Lattic Ĉ =(C,,,,, ) Data-Flow Functional [[ ]] : E (C C) Rvr Data-Flow Analyi Rvr Data-Flow Functional (Hugh, Launchbury 1992+) [[ ]] R : E (C C) dind by E c C. [[ ]] R (c)= d {c [[ ]](c ) c } Availability o Trm Abtract mantic or availability o trm 1. Data-Flow Lattic: (C,,,,, )= d (B,,,,al,tru) 2. Data-Flow Functional: [[ ]] av : E ( B B ) dind by Ct tru i Comp Tranp E. [[ ]] av = d Id B i Comp Tranp othrwi Ct al On th Rlationhip o [ ] and [ ] R Lmma 1. [[ ]] R i wll-dind and monotonic. 2. [[ ]] R i additiv, i [[ ]] i ditributiv. Monotonicity, Ditributivity, and Additivity o data-low unction. Dinition [Monotonicity, Ditributivity, Additivity] Lt Ĉ =(C,,,,, ) b a complt lattic and : C C a unction on C. Thn: i 1. monotonic i c,c C. c c (c) (c ) (Prrving th ordr o lmnt) 2. ditributiv i C C. ( C ) = {(c) c C } (Prrving gratt lowr bound) 3. additiv i C C. ( C ) = {(c) c C } (Prrving lat uppr bound) Otn uul th ollowing quivalnt charactrization o monotonicity: Lmma On th Rlationhip o [ ] and [ ] R (Cont d) Lmma 1. [[ ]] R [[ ]] Id C, i [[ ]] i monotonic. Lt Ĉ =(C,,,,, ) b a complt lattic and : C C a unction on C. Thn: i monotonic C C. ( C ) {(c) c C } ( C C. ( C ) {(c) c C } ) 2. [[ ]] [[ ]] R Id C, i [[ ]] i ditributiv. In trm o th thory o abtract intrprtation : [[ ]] and [[ ]] R orm a Galoi-connction
3 Rvr : Th R-MinFP -Approach Th R-MinFP -Equation Sytm: c q i n=q rqin (n)= {[[ (n,m) ]] R (rqin (m)) m ucc(n) } othrwi Th R-MinFP -Solution: c q C n N. R-MinFP cq (n)= d rqin c q (n) whr rqin c q dnot th lat olution o th R-MinFP -quation ytm wrt c q C. 17 Standard : Th MaxFP -Approach Th MaxFP -Equation Sytm: c i n= in (n) = {[[ (m,n) ]](in (m)) m prd(n) } othrwi Th MaxFP -Solution: c C n N. MaxFP ([[ ]],c)(n)= d in c (n) whr in c dnot th gratt olution o th MaxFP -quation ytm wrt c C. 18 Th Conncting Link Link Thorm For ditributiv data-low unctional [[ ]], q N, and c,c q C, w hav: R-MinFP cq () c MaxFP c (q) c q Continuing th Analogy o Standard and Rvr Data-Flow Analyi rgarding Soundn & Compltn (in trm o program vriication) / Saty & Coincidnc (Prciion) (in trm o data-low analyi) Ential th xtnibility o data-low unctional to path Id C i q <1 [[ p ]]= d [[ 2,, q ]] [[ 1 ]] othrwi Th MOP -Approach c C n N. MOP c (n)= {[[ p ]](c ) p P[,n] } Standard : Main Rult Thorm [Soundn / Saty] c C n N. MaxFP c (n) MOP c (n) i th data-low unctional [[ ]] i monotonic. Standard : Th Tool Kit Viw at a glanc: Intraprocdural C Spciication c 0 Thorm [Compltn / Coincidnc (Prciion)] Intraprocdural Framwork Thory Practic Tool Kit Gnric Fixd Point Alg. A.1 c C n N. MaxFP c (n)=mop c (n) i th data-low unctional [[ ]] i ditributiv. Program Proprty φ Equivalnc Intraprocdural Trmination Lmma Intraprocdural Intraprocdural Coincidnc Thorm Corrctn Lmma MOP-Solution MFP-Solution Computd Solution 1 2 3a) 3b) 23 24
4 O cour Rvr data-low unctional can b xtndd to path, too: Id C i q <1 [[ p ]] R = d [[ 1,, q 1 ]] R [[ q ]] R othrwi 25 Th R-JOP -Solution: Th R-JOP -Approach c q C n N. R-JOP cq (n)= d {[[ p ]] R (c q ) p P[n,q]} 26 Rvr : Main Rult Putting it togthr Thorm [Soundn / Rvr Saty] c q C n N. R-MinFP cq (n) R-JOP cq (n) Data-low Analyi PV {p}π {q} Program Vriication Thorm [Compltn / Rvr Coincidnc (Prciion)] MaxFP Coincidnc Thorm MOP (q) c c q C ditributiv induc C Strongt Potcondition Viw {p} π {?} c q C n N. R-MinFP cq (n)=r-jop cq (n) Link Thorm i [[ ]] i ditributiv. Rvr Coincidnc Thorm R-JOP R-MinFP c ( q ) c R (c) = d {c (c ) c} {?} π {q} Wakt Prcondition Viw Ar W Don? Rcall th motivating xampl A Poibly Hug Program Rgion Matring th Road to Succ rquir mor. It rquir u to conclud rom wakt pr-condition on trongt pot-condition. x ntially, thi man to rplac th analyi problm by a vriication problm. A Poibly Hug Program Rgion Changing th Prpctiv Implmntation Problm Spciication Problm Vriication Problm Changing th Prpctiv: Th Standard Taxonomy!cti?wci!cti Convntional Claiication o Tchniqu? ci! cpi! cpi! Givn: Contxt Inormation cti! Givn: Componnt Inormation cpi! Givn: Contxt Inormation cti Componnt Inormation cpi Exhautiv Dmand-Drivn? Sought: Strongt Componnt Inormation ci? Sought: Wakt Contxt Inormation wci? Sought: Validity o cpi with rpct o cti 31 32
5 Changing th Prpctiv: Concluion Drivd Th pciication problm: {?} π {q} Th vriication problm: {p} π {q}? th domain o dmand-drivn Th implmntation problm: {p} π {?} th domain o xhautiv 33 (R)-Framwork / (R)-Tool Kit Th Standard Viw Strongt Pot-condition Problm Partial (By-nd + Early Trmination) BN-MaxFP Exhautiv Program Analyi Formally anwr Formally anwr ( ) ( ) MaxFP MOP R-JOP R-MinFP 34 Wakt Pr-condition Problm Exhautiv Th Dual Viw Partial (Dmand-drivn + Early Trmination) DD-R-MinFP Gn/Kill-Problm allow u to matr th road to ucc: Th SPC-analyi problm boil down to a WPC-vriication problm. Thi i important bcau Rdundant Exprion/Aignmnt Elimination Dad-Cod Elimination Strngth Rduction Concluding th Exampl: Availability Abtract mantic or availability 1. Data-low lattic: (C,,,,, )= d (B X,,,,al,ailur) with =al tru ailur = 2. Data-low unctional: [[ ]] av : E ( B X B X ) dind by Ct X tru i Comp Tranp E. [[ ]] av = d Id BX i Comp Tranp othrwi Ct X al ar bad on Gn-Kill-problm Rvr Availability Rvr abtract mantic or availability 1. Data-low lattic: (C,,,,, )= d (B X,,,,al,ailur) 2. Rvr data-low unctional: [[ ]] avr : E ( B X B X ) dind by R-Ct X tru E. [[ ]] avr = d R-Id BX R-Ct X al i [[ ]] av =Ct X tru i [[ ]] av =Id BX i [[ ]] av =Ct X al Supporting Function al b B X. R-Ct X tru(b) = d ailur b B X. R-Ct X al al(b) = d ailur R-Id BX = d Id BX i b B othrwi (i.., i b=ailur) i b=al othrwi Summing Up / Extnion (R)-Framwork / (R)-Tool Kit (Cont d) In thi talk th gnral pattrn: Th intraprocdural baic tting o (R) (Knoop, KPS 2007) Spciication Extnion ar poibl Framwork Thory Practic Intrac Tool Kit Intrprocdural tting (Knoop, CC 1992, LNCS 1428 (1998)) Gnric Fixd Point Alg. Paralll tting (Knoop, Euro-Par 1999) Program Proprty φ Equivalnc MOP-Solution Coincidnc Thorm MFP-Solution Corrctn Lmma Trmination Computd Solution 1 2 3a) 3b) Lmma 39 40
6 (R)-Framwork / (R)-Tool Kit th gnral pattrn mor abtract: Program Proprty φ Proo Obligation: Intraprocdural Intrprocdural Conditional Spciication Ectivity Coincidnc Thorm Thorm 3a) 3b) 1 2 3a) 3b) Equivalnc Paralll Coincidnc Framwork 41 Corrctn Trmination Ectivity From Application toward Concluion a) c := a+b rad(a,b) x a := "Hot Spot" Optimizr Program point atii availability, whil do not! Application b := writ(c) b) 42 c := a+b rad(a,b) a := Dbuggr b := writ(c) Variabl c i not initializd along om path raching program point g Rvr Data-Flow Analyi pcially wll-uitd or Hot-Spot Optimization Dbugging Jut-in-tim Compilation Concluion (Cont d) bad on anwring data-low quri. Hnc A an appaling add-on R i tailord or paralllization! Data-Flow Analyi or Dbugging Data-Flow Analyi or Jut-in-tim Compilation wr titl conidrd optionally Rcall again A Poibly Hug Program Rgion Concluion and Prpctiv x Data-Flow Analyi or Multi-Cor Architctur A Poibly Hug Program Rgion 45 46
Lecture 4: Parsing. Administrivia
Adminitrivia Lctur 4: Paring If you do not hav a group, pla pot a rqut on Piazzza ( th Form projct tam... itm. B ur to updat your pot if you find on. W will aign orphan to group randomly in a fw day. Programming
More informationSource code. where each α ij is a terminal or nonterminal symbol. We say that. α 1 α m 1 Bα m+1 α n α 1 α m 1 β 1 β p α m+1 α n
Adminitrivia Lctur : Paring If you do not hav a group, pla pot a rqut on Piazzza ( th Form projct tam... itm. B ur to updat your pot if you find on. W will aign orphan to group randomly in a fw day. Programming
More information[ ] 1+ lim G( s) 1+ s + s G s s G s Kacc SYSTEM PERFORMANCE. Since. Lecture 10: Steady-state Errors. Steady-state Errors. Then
SYSTEM PERFORMANCE Lctur 0: Stady-tat Error Stady-tat Error Lctur 0: Stady-tat Error Dr.alyana Vluvolu Stady-tat rror can b found by applying th final valu thorm and i givn by lim ( t) lim E ( ) t 0 providd
More informationECE Spring Prof. David R. Jackson ECE Dept. Notes 6
ECE 6345 Spring 2015 Prof. David R. Jackon ECE Dpt. Not 6 1 Ovrviw In thi t of not w look at two diffrnt modl for calculating th radiation pattrn of a microtrip antnna: Elctric currnt modl Magntic currnt
More informationSection 11.6: Directional Derivatives and the Gradient Vector
Sction.6: Dirctional Drivativs and th Gradint Vctor Practic HW rom Stwart Ttbook not to hand in p. 778 # -4 p. 799 # 4-5 7 9 9 35 37 odd Th Dirctional Drivativ Rcall that a b Slop o th tangnt lin to th
More informationNEW APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA
NE APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA Mirca I CÎRNU Ph Dp o Mathmatics III Faculty o Applid Scincs Univrsity Polithnica o Bucharst Cirnumirca @yahoocom Abstract In a rcnt papr [] 5 th indinit intgrals
More informationAbstract Interpretation: concrete and abstract semantics
Abstract Intrprtation: concrt and abstract smantics Concrt smantics W considr a vry tiny languag that manags arithmtic oprations on intgrs valus. Th (concrt) smantics of th languags cab b dfind by th funzcion
More informationcycle that does not cross any edges (including its own), then it has at least
W prov th following thorm: Thorm If a K n is drawn in th plan in such a way that it has a hamiltonian cycl that dos not cross any dgs (including its own, thn it has at last n ( 4 48 π + O(n crossings Th
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 301 Signals & Systms Prof. Mark Fowlr ot St #21 D-T Signals: Rlation btwn DFT, DTFT, & CTFT 1/16 W can us th DFT to implmnt numrical FT procssing This nabls us to numrically analyz a signal to find
More informationSupplementary Materials
6 Supplmntary Matrials APPENDIX A PHYSICAL INTERPRETATION OF FUEL-RATE-SPEED FUNCTION A truck running on a road with grad/slop θ positiv if moving up and ngativ if moving down facs thr rsistancs: arodynamic
More informationpriority queue ADT heaps 1
COMP 250 Lctur 23 priority quu ADT haps 1 Nov. 1/2, 2017 1 Priority Quu Li a quu, but now w hav a mor gnral dinition o which lmnt to rmov nxt, namly th on with highst priority..g. hospital mrgncy room
More informationRecall that by Theorems 10.3 and 10.4 together provide us the estimate o(n2 ), S(q) q 9, q=1
Chaptr 11 Th singular sris Rcall that by Thorms 10 and 104 togthr provid us th stimat 9 4 n 2 111 Rn = SnΓ 2 + on2, whr th singular sris Sn was dfind in Chaptr 10 as Sn = q=1 Sq q 9, with Sq = 1 a q gcda,q=1
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!
More informationAbstract Interpretation. Lecture 5. Profs. Aiken, Barrett & Dill CS 357 Lecture 5 1
Abstract Intrprtation 1 History On brakthrough papr Cousot & Cousot 77 (?) Inspird by Dataflow analysis Dnotational smantics Enthusiastically mbracd by th community At last th functional community... At
More informationPartial Derivatives: Suppose that z = f(x, y) is a function of two variables.
Chaptr Functions o Two Variabls Applid Calculus 61 Sction : Calculus o Functions o Two Variabls Now that ou hav som amiliarit with unctions o two variabls it s tim to start appling calculus to hlp us solv
More information(1) Then we could wave our hands over this and it would become:
MAT* K285 Spring 28 Anthony Bnoit 4/17/28 Wk 12: Laplac Tranform Rading: Kohlr & Johnon, Chaptr 5 to p. 35 HW: 5.1: 3, 7, 1*, 19 5.2: 1, 5*, 13*, 19, 45* 5.3: 1, 11*, 19 * Pla writ-up th problm natly and
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!
More informationA Uniform Approach to Three-Valued Semantics for µ-calculus on Abstractions of Hybrid Automata
A Uniform Approach to Thr-Valud Smantics for µ-calculus on Abstractions of Hybrid Automata (Haifa Vrification Confrnc 2008) Univrsity of Kaisrslautrn Octobr 28, 2008 Ovrviw 1. Prliminaris and 2. Gnric
More informationBasic Polyhedral theory
Basic Polyhdral thory Th st P = { A b} is calld a polyhdron. Lmma 1. Eithr th systm A = b, b 0, 0 has a solution or thr is a vctorπ such that π A 0, πb < 0 Thr cass, if solution in top row dos not ist
More informationAddition of angular momentum
Addition of angular momntum April, 07 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat
More informationInjective topological fibre spaces
Topology and its pplications 125 (2002) 525 532 www.lsvir.com/locat/topol Injctiv topological ibr spacs F. Cagliari a,,s.mantovani b a Dipartimnto di Matmatica, Univrsità di Bologna, Piazza di Porta S.
More informationChapter 10. The singular integral Introducing S(n) and J(n)
Chaptr Th singular intgral Our aim in this chaptr is to rplac th functions S (n) and J (n) by mor convnint xprssions; ths will b calld th singular sris S(n) and th singular intgral J(n). This will b don
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Procssing Prof. Mark Fowlr ot St #18 Introduction to DFT (via th DTFT) Rading Assignmnt: Sct. 7.1 of Proakis & Manolakis 1/24 Discrt Fourir Transform (DFT) W v sn that th DTFT is
More informationSECTION where P (cos θ, sin θ) and Q(cos θ, sin θ) are polynomials in cos θ and sin θ, provided Q is never equal to zero.
SETION 6. 57 6. Evaluation of Dfinit Intgrals Exampl 6.6 W hav usd dfinit intgrals to valuat contour intgrals. It may com as a surpris to larn that contour intgrals and rsidus can b usd to valuat crtain
More informationLESSON 10: THE LAPLACE TRANSFORM
0//06 lon0t438a.pptx ESSON 0: THE APAE TANSFOM ET 438a Automatic ontrol Sytm Tchnology arning Objctiv Aftr thi prntation you will b abl to: Explain how th aplac tranform rlat to th tranint and inuoidal
More informationMultiple Short Term Infusion Homework # 5 PHA 5127
Multipl Short rm Infusion Homwork # 5 PHA 527 A rug is aministr as a short trm infusion. h avrag pharmacokintic paramtrs for this rug ar: k 0.40 hr - V 28 L his rug follows a on-compartmnt boy mol. A 300
More informationBINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES
BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES DONALD M. DAVIS Abstract. If p is a prim (implicit in notation and n a positiv intgr, lt ν(n dnot th xponnt of p in n, and U(n n/p ν(n, th unit
More informationPropositional Logic. Combinatorial Problem Solving (CPS) Albert Oliveras Enric Rodríguez-Carbonell. May 17, 2018
Propositional Logic Combinatorial Problm Solving (CPS) Albrt Olivras Enric Rodríguz-Carbonll May 17, 2018 Ovrviw of th sssion Dfinition of Propositional Logic Gnral Concpts in Logic Rduction to SAT CNFs
More informationAddition of angular momentum
Addition of angular momntum April, 0 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat th
More informationStrongly Connected Components
Strongly Connctd Componnts Lt G = (V, E) b a dirctd graph Writ if thr is a path from to in G Writ if and is an quivalnc rlation: implis and implis s quivalnc classs ar calld th strongly connctd componnts
More informationFunction Spaces. a x 3. (Letting x = 1 =)) a(0) + b + c (1) = 0. Row reducing the matrix. b 1. e 4 3. e 9. >: (x = 1 =)) a(0) + b + c (1) = 0
unction Spacs Prrquisit: Sction 4.7, Coordinatization n this sction, w apply th tchniqus of Chaptr 4 to vctor spacs whos lmnts ar functions. Th vctor spacs P n and P ar familiar xampls of such spacs. Othr
More informationBINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES. 1. Statement of results
BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES DONALD M. DAVIS Abstract. If p is a prim and n a positiv intgr, lt ν p (n dnot th xponnt of p in n, and u p (n n/p νp(n th unit part of n. If α
More informationFourier Transforms and the Wave Equation. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation.
Lur 7 Fourir Transforms and th Wav Euation Ovrviw and Motivation: W first discuss a fw faturs of th Fourir transform (FT), and thn w solv th initial-valu problm for th wav uation using th Fourir transform
More informationIntegration Continued. Integration by Parts Solving Definite Integrals: Area Under a Curve Improper Integrals
Intgrtion Continud Intgrtion y Prts Solving Dinit Intgrls: Ar Undr Curv Impropr Intgrls Intgrtion y Prts Prticulrly usul whn you r trying to tk th intgrl o som unction tht is th product o n lgric prssion
More informationSection 6.1. Question: 2. Let H be a subgroup of a group G. Then H operates on G by left multiplication. Describe the orbits for this operation.
MAT 444 H Barclo Spring 004 Homwork 6 Solutions Sction 6 Lt H b a subgroup of a group G Thn H oprats on G by lft multiplication Dscrib th orbits for this opration Th orbits of G ar th right costs of H
More informationPROBLEM SET Problem 1.
PROLEM SET 1 PROFESSOR PETER JOHNSTONE 1. Problm 1. 1.1. Th catgory Mat L. OK, I m not amiliar with th trminology o partially orr sts, so lt s go ovr that irst. Dinition 1.1. partial orr is a binary rlation
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Procssing Prof. Mark Fowlr Dtails of th ot St #19 Rading Assignmnt: Sct. 7.1.2, 7.1.3, & 7.2 of Proakis & Manolakis Dfinition of th So Givn signal data points x[n] for n = 0,, -1
More informationY 0. Standing Wave Interference between the incident & reflected waves Standing wave. A string with one end fixed on a wall
Staning Wav Intrfrnc btwn th incint & rflct wavs Staning wav A string with on n fix on a wall Incint: y, t) Y cos( t ) 1( Y 1 ( ) Y (St th incint wav s phas to b, i.., Y + ral & positiv.) Rflct: y, t)
More informationSCHUR S THEOREM REU SUMMER 2005
SCHUR S THEOREM REU SUMMER 2005 1. Combinatorial aroach Prhas th first rsult in th subjct blongs to I. Schur and dats back to 1916. On of his motivation was to study th local vrsion of th famous quation
More informationLecture 37 (Schrödinger Equation) Physics Spring 2018 Douglas Fields
Lctur 37 (Schrödingr Equation) Physics 6-01 Spring 018 Douglas Filds Rducd Mass OK, so th Bohr modl of th atom givs nrgy lvls: E n 1 k m n 4 But, this has on problm it was dvlopd assuming th acclration
More informationEngineering Differential Equations Practice Final Exam Solutions Fall 2011
9.6 Enginring Diffrntial Equation Practic Final Exam Solution Fall 0 Problm. (0 pt.) Solv th following initial valu problm: x y = xy, y() = 4. Thi i a linar d.. bcau y and y appar only to th firt powr.
More information10. The Discrete-Time Fourier Transform (DTFT)
Th Discrt-Tim Fourir Transform (DTFT Dfinition of th discrt-tim Fourir transform Th Fourir rprsntation of signals plays an important rol in both continuous and discrt signal procssing In this sction w
More informationDeift/Zhou Steepest descent, Part I
Lctur 9 Dift/Zhou Stpst dscnt, Part I W now focus on th cas of orthogonal polynomials for th wight w(x) = NV (x), V (x) = t x2 2 + x4 4. Sinc th wight dpnds on th paramtr N N w will writ π n,n, a n,n,
More informationFrom Elimination to Belief Propagation
School of omputr Scinc Th lif Propagation (Sum-Product lgorithm Probabilistic Graphical Modls (10-708 Lctur 5, Sp 31, 2007 Rcptor Kinas Rcptor Kinas Kinas X 5 ric Xing Gn G T X 6 X 7 Gn H X 8 Rading: J-hap
More informationFirst derivative analysis
Robrto s Nots on Dirntial Calculus Chaptr 8: Graphical analysis Sction First drivativ analysis What you nd to know alrady: How to us drivativs to idntiy th critical valus o a unction and its trm points
More informationINTEGRATION BY PARTS
Mathmatics Rvision Guids Intgration by Parts Pag of 7 MK HOME TUITION Mathmatics Rvision Guids Lvl: AS / A Lvl AQA : C Edcl: C OCR: C OCR MEI: C INTEGRATION BY PARTS Vrsion : Dat: --5 Eampls - 6 ar copyrightd
More informationChapter 6 Folding. Folding
Chaptr 6 Folding Wintr 1 Mokhtar Abolaz Folding Th folding transformation is usd to systmatically dtrmin th control circuits in DSP architctur whr multipl algorithm oprations ar tim-multiplxd to a singl
More informationLINEAR DELAY DIFFERENTIAL EQUATION WITH A POSITIVE AND A NEGATIVE TERM
Elctronic Journal of Diffrntial Equations, Vol. 2003(2003), No. 92, pp. 1 6. ISSN: 1072-6691. URL: http://jd.math.swt.du or http://jd.math.unt.du ftp jd.math.swt.du (login: ftp) LINEAR DELAY DIFFERENTIAL
More informationLenses & Prism Consider light entering a prism At the plane surface perpendicular light is unrefracted Moving from the glass to the slope side light
Lnss & Prism Considr light ntring a prism At th plan surac prpndicular light is unrractd Moving rom th glass to th slop sid light is bnt away rom th normal o th slop Using Snll's law n sin( ϕ ) = n sin(
More informationRELATIONS BETWEEN GABOR TRANSFORMS AND FRACTIONAL FOURIER TRANSFORMS AND THEIR APPLICATIONS FOR SIGNAL PROCESSING
RELATIONS BETWEEN ABOR TRANSFORMS AND FRACTIONAL FOURIER TRANSFORMS AND THEIR APPLICATIONS FOR SINAL PROCESSIN Soo-Chang Pi, Jian-Jiun Ding Dpartmnt o Elctrical Enginring, National Taiwan Univrsity, No.,
More informationNetwork Congestion Games
Ntwork Congstion Gams Assistant Profssor Tas A&M Univrsity Collg Station, TX TX Dallas Collg Station Austin Houston Bst rout dpnds on othrs Ntwork Congstion Gams Travl tim incrass with congstion Highway
More informationBifurcation Theory. , a stationary point, depends on the value of α. At certain values
Dnamic Macroconomic Thor Prof. Thomas Lux Bifurcation Thor Bifurcation: qualitativ chang in th natur of th solution occurs if a paramtr passs through a critical point bifurcation or branch valu. Local
More informationCPSC 665 : An Algorithmist s Toolkit Lecture 4 : 21 Jan Linear Programming
CPSC 665 : An Algorithmist s Toolkit Lctur 4 : 21 Jan 2015 Lcturr: Sushant Sachdva Linar Programming Scrib: Rasmus Kyng 1. Introduction An optimization problm rquirs us to find th minimum or maximum) of
More informationThat is, we start with a general matrix: And end with a simpler matrix:
DIAGON ALIZATION OF THE STR ESS TEN SOR INTRO DUCTIO N By th us of Cauchy s thorm w ar abl to rduc th numbr of strss componnts in th strss tnsor to only nin valus. An additional simplification of th strss
More informationAPPLICATIONS OF THE LAPLACE-MELLIN INTEGRAL TRANSFORM TO DIFFERNTIAL EQUATIONS
Intrntionl Journl o Sintii nd Rrh Publition Volum, Iu 5, M ISSN 5-353 APPLICATIONS OF THE LAPLACE-MELLIN INTEGRAL TRANSFORM TO DIFFERNTIAL EQUATIONS S.M.Khirnr, R.M.Pi*, J.N.Slun** Dprtmnt o Mthmti Mhrhtr
More informationThe Equitable Dominating Graph
Intrnational Journal of Enginring Rsarch and Tchnology. ISSN 0974-3154 Volum 8, Numbr 1 (015), pp. 35-4 Intrnational Rsarch Publication Hous http://www.irphous.com Th Equitabl Dominating Graph P.N. Vinay
More informationExercise 1. Sketch the graph of the following function. (x 2
Writtn tst: Fbruary 9th, 06 Exrcis. Sktch th graph of th following function fx = x + x, spcifying: domain, possibl asymptots, monotonicity, continuity, local and global maxima or minima, and non-drivability
More informationMCE503: Modeling and Simulation of Mechatronic Systems Discussion on Bond Graph Sign Conventions for Electrical Systems
MCE503: Modling and Simulation o Mchatronic Systms Discussion on Bond Graph Sign Convntions or Elctrical Systms Hanz ichtr, PhD Clvland Stat Univrsity, Dpt o Mchanical Enginring 1 Basic Assumption In a
More informationProblem Set 6 Solutions
6.04/18.06J Mathmatics for Computr Scinc March 15, 005 Srini Dvadas and Eric Lhman Problm St 6 Solutions Du: Monday, March 8 at 9 PM in Room 3-044 Problm 1. Sammy th Shark is a financial srvic providr
More informationEinstein Equations for Tetrad Fields
Apiron, Vol 13, No, Octobr 006 6 Einstin Equations for Ttrad Filds Ali Rıza ŞAHİN, R T L Istanbul (Turky) Evry mtric tnsor can b xprssd by th innr product of ttrad filds W prov that Einstin quations for
More informationME 321 Kinematics and Dynamics of Machines S. Lambert Winter 2002
3.4 Forc Analysis of Linkas An undrstandin of forc analysis of linkas is rquird to: Dtrmin th raction forcs on pins, tc. as a consqunc of a spcifid motion (don t undrstimat th sinificanc of dynamic or
More informationu 3 = u 3 (x 1, x 2, x 3 )
Lctur 23: Curvilinar Coordinats (RHB 8.0 It is oftn convnint to work with variabls othr than th Cartsian coordinats x i ( = x, y, z. For xampl in Lctur 5 w mt sphrical polar and cylindrical polar coordinats.
More informationThe Frequency Response of a Quarter-Wave Matching Network
4/1/29 Th Frquncy Rsons o a Quartr 1/9 Th Frquncy Rsons o a Quartr-Wav Matchg Ntwork Q: You hav onc aga rovidd us with conusg and rhas uslss ormation. Th quartr-wav matchg ntwork has an xact SFG o: a Τ
More informationLecture 6.4: Galois groups
Lctur 6.4: Galois groups Matthw Macauly Dpartmnt of Mathmatical Scincs Clmson Univrsity http://www.math.clmson.du/~macaul/ Math 4120, Modrn Algbra M. Macauly (Clmson) Lctur 6.4: Galois groups Math 4120,
More informationData Assimilation 1. Alan O Neill National Centre for Earth Observation UK
Data Assimilation 1 Alan O Nill National Cntr for Earth Obsrvation UK Plan Motivation & basic idas Univariat (scalar) data assimilation Multivariat (vctor) data assimilation 3d-Variational Mthod (& optimal
More informationComputing and Communications -- Network Coding
89 90 98 00 Computing and Communications -- Ntwork Coding Dr. Zhiyong Chn Institut of Wirlss Communications Tchnology Shanghai Jiao Tong Univrsity China Lctur 5- Nov. 05 0 Classical Information Thory Sourc
More informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #20 11/14/2013
18.782 Introduction to Arithmtic Gomtry Fall 2013 Lctur #20 11/14/2013 20.1 Dgr thorm for morphisms of curvs Lt us rstat th thorm givn at th nd of th last lctur, which w will now prov. Thorm 20.1. Lt φ:
More informationELECTRON-MUON SCATTERING
ELECTRON-MUON SCATTERING ABSTRACT Th lctron charg is considrd to b distributd or xtndd in spac. Th diffrntial of th lctron charg is st qual to a function of lctron charg coordinats multiplid by a four-dimnsional
More informationANALYSIS IN THE FREQUENCY DOMAIN
ANALYSIS IN THE FREQUENCY DOMAIN SPECTRAL DENSITY Dfinition Th spctral dnsit of a S.S.P. t also calld th spctrum of t is dfind as: + { γ }. jτ γ τ F τ τ In othr words, of th covarianc function. is dfind
More informationPipe flow friction, small vs. big pipes
Friction actor (t/0 t o pip) Friction small vs larg pips J. Chaurtt May 016 It is an intrsting act that riction is highr in small pips than largr pips or th sam vlocity o low and th sam lngth. Friction
More informationSearching Linked Lists. Perfect Skip List. Building a Skip List. Skip List Analysis (1) Assume the list is sorted, but is stored in a linked list.
3 3 4 8 6 3 3 4 8 6 3 3 4 8 6 () (d) 3 Sarching Linkd Lists Sarching Linkd Lists Sarching Linkd Lists ssum th list is sortd, but is stord in a linkd list. an w us binary sarch? omparisons? Work? What if
More informationFirst order differential equation Linear equation; Method of integrating factors
First orr iffrntial quation Linar quation; Mtho of intgrating factors Exampl 1: Rwrit th lft han si as th rivativ of th prouct of y an som function by prouct rul irctly. Solving th first orr iffrntial
More informationDISTRIBUTION OF DIFFERENCE BETWEEN INVERSES OF CONSECUTIVE INTEGERS MODULO P
DISTRIBUTION OF DIFFERENCE BETWEEN INVERSES OF CONSECUTIVE INTEGERS MODULO P Tsz Ho Chan Dartmnt of Mathmatics, Cas Wstrn Rsrv Univrsity, Clvland, OH 4406, USA txc50@cwru.du Rcivd: /9/03, Rvisd: /9/04,
More information( ) = ( ) ( ) ( ) ( ) τ τ. This is a more complete version of the solutions for assignment 2 courtesy of the course TA
This is a mor complt vrsion o th solutions or assignmnt courtsy o th cours TA ) Find th Fourir transorms o th signals shown in Figur (a-b). -a) -b) From igur -a, w hav: g t 4 0 t = t 0 othrwis = j G g
More informationWeek 3: Connected Subgraphs
Wk 3: Connctd Subgraphs Sptmbr 19, 2016 1 Connctd Graphs Path, Distanc: A path from a vrtx x to a vrtx y in a graph G is rfrrd to an xy-path. Lt X, Y V (G). An (X, Y )-path is an xy-path with x X and y
More informationBSc Engineering Sciences A. Y. 2017/18 Written exam of the course Mathematical Analysis 2 August 30, x n, ) n 2
BSc Enginring Scincs A. Y. 27/8 Writtn xam of th cours Mathmatical Analysis 2 August, 28. Givn th powr sris + n + n 2 x n, n n dtrmin its radius of convrgnc r, and study th convrgnc for x ±r. By th root
More informationMATH 319, WEEK 15: The Fundamental Matrix, Non-Homogeneous Systems of Differential Equations
MATH 39, WEEK 5: Th Fundamntal Matrix, Non-Homognous Systms of Diffrntial Equations Fundamntal Matrics Considr th problm of dtrmining th particular solution for an nsmbl of initial conditions For instanc,
More informationOn spanning trees and cycles of multicolored point sets with few intersections
On spanning trs and cycls of multicolord point sts with fw intrsctions M. Kano, C. Mrino, and J. Urrutia April, 00 Abstract Lt P 1,..., P k b a collction of disjoint point sts in R in gnral position. W
More informationUnit 6: Solving Exponential Equations and More
Habrman MTH 111 Sction II: Eonntial and Logarithmic Functions Unit 6: Solving Eonntial Equations and Mor EXAMPLE: Solv th quation 10 100 for. Obtain an act solution. This quation is so asy to solv that
More informationCOUNTING TAMELY RAMIFIED EXTENSIONS OF LOCAL FIELDS UP TO ISOMORPHISM
COUNTING TAMELY RAMIFIED EXTENSIONS OF LOCAL FIELDS UP TO ISOMORPHISM Jim Brown Dpartmnt of Mathmatical Scincs, Clmson Univrsity, Clmson, SC 9634, USA jimlb@g.clmson.du Robrt Cass Dpartmnt of Mathmatics,
More informationDirect Approach for Discrete Systems One-Dimensional Elements
CONTINUUM & FINITE ELEMENT METHOD Dirct Approach or Discrt Systms On-Dimnsional Elmnts Pro. Song Jin Par Mchanical Enginring, POSTECH Dirct Approach or Discrt Systms Dirct approach has th ollowing aturs:
More informationBasic Logic Review. Rules. Lecture Roadmap Combinational Logic. Textbook References. Basic Logic Gates (2-input versions)
Lctur Roadmap ombinational Logic EE 55 Digital Systm Dsign with VHDL Lctur Digital Logic Rrshr Part ombinational Logic Building Blocks Basic Logic Rviw Basic Gats D Morgan s Law ombinational Logic Building
More informationINTRODUCTION TO AUTOMATIC CONTROLS INDEX LAPLACE TRANSFORMS
adjoint...6 block diagram...4 clod loop ytm... 5, 0 E()...6 (t)...6 rror tady tat tracking...6 tracking...6...6 gloary... 0 impul function...3 input...5 invr Laplac tranform, INTRODUCTION TO AUTOMATIC
More informationSome remarks on Kurepa s left factorial
Som rmarks on Kurpa s lft factorial arxiv:math/0410477v1 [math.nt] 21 Oct 2004 Brnd C. Kllnr Abstract W stablish a connction btwn th subfactorial function S(n) and th lft factorial function of Kurpa K(n).
More informationHigher-Order Discrete Calculus Methods
Highr-Ordr Discrt Calculus Mthods J. Blair Prot V. Subramanian Ralistic Practical, Cost-ctiv, Physically Accurat Paralll, Moving Msh, Complx Gomtry, Slid 1 Contxt Discrt Calculus Mthods Finit Dirnc Mimtic
More informationCOHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim.
MTH rviw part b Lucian Mitroiu Th LOG and EXP functions Th ponntial function p : R, dfind as Proprtis: lim > lim p Eponntial function Y 8 6 - -8-6 - - X Th natural logarithm function ln in US- log: function
More informationENGR 323 BHW 15 Van Bonn 1/7
ENGR 33 BHW 5 Van Bonn /7 4.4 In Eriss and 3 as wll as man othr situations on has th PDF o X and wishs th PDF o Yh. Assum that h is an invrtibl untion so that h an b solvd or to ild. Thn it an b shown
More information1997 AP Calculus AB: Section I, Part A
997 AP Calculus AB: Sction I, Part A 50 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr.. (4 6 ) d= 4 6 6
More informationFigure 1: Closed surface, surface with boundary, or not a surface?
QUESTION 1 (10 marks) Two o th topological spacs shown in Figur 1 ar closd suracs, two ar suracs with boundary, and two ar not suracs. Dtrmin which is which. You ar not rquird to justiy your answr, but,
More informationCharacterizations of Continuous Distributions by Truncated Moment
Journal o Modrn Applid Statistical Mthods Volum 15 Issu 1 Articl 17 5-016 Charactrizations o Continuous Distributions by Truncatd Momnt M Ahsanullah Ridr Univrsity M Shakil Miami Dad Coll B M Golam Kibria
More informationDISCRETE TIME FOURIER TRANSFORM (DTFT)
DISCRETE TIME FOURIER TRANSFORM (DTFT) Th dicrt-tim Fourir Tranform x x n xn n n Th Invr dicrt-tim Fourir Tranform (IDTFT) x n Not: ( ) i a complx valud continuou function = π f [rad/c] f i th digital
More informationRoadmap. XML Indexing. DataGuide example. DataGuides. Strong DataGuides. Multiple DataGuides for same data. CPS Topics in Database Systems
Roadmap XML Indxing CPS 296.1 Topics in Databas Systms Indx fabric Coopr t al. A Fast Indx for Smistructurd Data. VLDB, 2001 DataGuid Goldman and Widom. DataGuids: Enabling Qury Formulation and Optimization
More informationWhat is a hereditary algebra?
What is a hrditary algbra? (On Ext 2 and th vanishing of Ext 2 ) Claus Michal Ringl At th Münstr workshop 2011, thr short lcturs wr arrangd in th styl of th rgular column in th Notics of th AMS: What is?
More informationON A CONJECTURE OF RYSElt AND MINC
MA THEMATICS ON A CONJECTURE OF RYSElt AND MINC BY ALBERT NIJE~HUIS*) AND HERBERT S. WILF *) (Communicatd at th mting of January 31, 1970) 1. Introduction Lt A b an n x n matrix of zros and ons, and suppos
More informationQuantum Phase Operator and Phase States
Quantum Pha Oprator and Pha Stat Xin Ma CVS Halth Richardon Txa 75081 USA William Rhod Dpartmnt of Chmitry Florida Stat Univrity Tallaha Florida 3306 USA A impl olution i prntd to th long-tanding Dirac
More informationDerivation of Eigenvalue Matrix Equations
Drivation of Eignvalu Matrix Equations h scalar wav quations ar φ φ η + ( k + 0ξ η β ) φ 0 x y x pq ε r r whr for E mod E, 1, y pq φ φ x 1 1 ε r nr (4 36) for E mod H,, 1 x η η ξ ξ n [ N ] { } i i i 1
More informationConstruction of asymmetric orthogonal arrays of strength three via a replacement method
isid/ms/26/2 Fbruary, 26 http://www.isid.ac.in/ statmath/indx.php?modul=prprint Construction of asymmtric orthogonal arrays of strngth thr via a rplacmnt mthod Tian-fang Zhang, Qiaoling Dng and Alok Dy
More informationAn Application of Hardy-Littlewood Conjecture. JinHua Fei. ChangLing Company of Electronic Technology Baoji Shannxi P.R.China
An Application of Hardy-Littlwood Conjctur JinHua Fi ChangLing Company of Elctronic Tchnology Baoji Shannxi P.R.China E-mail: fijinhuayoujian@msn.com Abstract. In this papr, w assum that wakr Hardy-Littlwood
More informationExponential Functions
Eponntial Functions Dinition: An Eponntial Function is an unction tat as t orm a, wr a > 0. T numbr a is calld t bas. Eampl: Lt i.. at intgrs. It is clar wat t unction mans or som valus o. 0 0,,, 8,,.,.
More informationCLONES IN 3-CONNECTED FRAME MATROIDS
CLONES IN 3-CONNECTED FRAME MATROIDS JAKAYLA ROBBINS, DANIEL SLILATY, AND XIANGQIAN ZHOU Abstract. W dtrmin th structur o clonal classs o 3-connctd ram matroids in trms o th structur o biasd graphs. Robbins
More information