Causality Consistency Problem for Two Different Possibilistic Causal Models
|
|
- Clemence Fletcher
- 5 years ago
- Views:
Transcription
1 nernaonal ymoum on Medcal nformac and Fuzzy Technoloy (MF Hano (Au. 999 Caualy Conency Problem for Two Dfferen Poblc Caual Model Koch YAMADA Dearmen of Plannn and Manaemen cence Naaoka Unvery of Technoloy 603- Kam-omoka Naaoka Naa JAPAN E-mal: ABTRACT Condonal Caual Probably (CCPR and Condonal Caual Pobly (CCPO have been rooed o exre exac uncerane of cauale and ome reaonn mehod baed on hem have been uded o calculae robable or oble of unknown even under he condon ha ome even are known. CCPR/CCPO a condonal robably/obly of a cauaon even condoned by caue. Cauaon even an Òeven ha a caue acually caue an effec.ó The aer rooe o clafy caual model baed on cauaon even no wo ye -- ymmercally valued and aymmercally valued caual model -- deendn on he roere of varable whch ake an even a her value. alo how he relaon beween CCPO and convenonal condonal oble n hee wo ye of caual model. Then defne and dcue a Caualy Conency Problem whch a roblem o calculae he obly of combned value of arbrarly choen unknown varable when value of ome oher varable are known.. NTRODUCTON Condonal Caual Probably (CCPR and Condonal Caual Pobly (CCPO have been rooed o exre exac uncerane of cauale and ome reaonn mehod baed on hem have been uded o calculae uncerane of unknown even when ome oher even are known [-5]. CCPR defned a condonal robably of a cauaon even condoned by caue. Cauaon even an "even ha a caue are and acually caue an effec [-3]." CCPO a oblc veron of he dea [45] whch emloy Pobly heory [67] o exre uncerane nead of Probably heory. The aer fr rooe o clafy caual model baed on cauaon even no wo ye -- ymmercally valued and aymmercally valued caual model -- deendn on he roere of varable whch ake an even a her value. alo how ha he relaon beween CCPO and convenonal condonal oble are dfferen beween hee wo ye of model. Then defne a Caualy Conency Problem and dcue oluon and her roere. The defned roblem one o calculae he obly of combned value of arbrarly choen unknown varable when value of ome oher varable are known. The roblem an exended one ncludn nvere roblem of caualy [4] and caualy analy roblem [5] uded n he revou reearch. Wha hould be noced n he dcuon for he roere of oluon ha we do no have o be cauou wh he dfference of wo caual model. Th becaue he dcuon oble un convenonal condonal oble derved from CCPO exce a roof of a rooon. A he end of he aer a numercal examle of a ymmercally valued model hown. The examle verfe ome of he roere of oluon dcued n he aer. 2. TWO DFFERENT CAUAL MODEL Fr le u defne caue effec and cauaon even whch were nroduced n [-3]. [Defnon 2.] Le x (É and y (=ÉJ be varable akn a value n a e of even U ={u Éu H and V ={v Év K reecvely. The value of y deenden on hoe of ome x. Thu u h (h=éh and v k (k=ék are called caue and effec even reecvely. x ( y a caue (an effec varable. Then cauaon even one ha Òu h are and u h acually caue v k Ó and exreed n v k :u h 82
2 afyn he follown equaon: v : u «( v : u u «( v : u v «( v : u u v k h k h h k h k k h h k ( v : u u «u (2 k h h h v : u v «v (3 k h k k where «and mean equvalence and local roduc. u h denoe he abence of u h. When v k :u h rue ad ha u h uor v k. Then we defne he ymmercally valued caual model menoned n NTRODUCTON. [Defnon 2.2] When x Î{ u... uh and y Î{ v... vk afy he nex equaon x and y comoe a ymmercally valued caual (VC model. vk «( vk : uh. (4 h The above defnon how ha an effec even are f and only f caued by one or more of caue even. Th called Mandaory Cauaon Aumon n [5] houh he word wa ued a fr n aymmercally valued caual model a decrbed below [3]. Now we conder a cae where H=K=2 ha U ={u u 2 and V ={v v 2. f we aume u = u and v = v we can rewre U = { u u and V = { v v becaue U and V are excluve and exhauve e. Then he aymmercally valued caual model defned below. [Defnon 2.3] When x Î{ u uand y Î{ v v afy he follown equaon x and y comoe an aymmercally valued caual (AVC model. v «( v : u. (5 v : u «v : u «v : u «F (6 where F mean fale. n he above eq. (6 mean ha u canno uor any v and ha v canno be uored by any caue even. Therefore obvou ha eq. (4 doe no hold for v = v 2. nead we e he nex from eq. (5. v «( v : u (7 For an AVC model Mandaory Cauaon Aumon mean ha eq. (5 hold [3]. The dfference beween a VC and an AVC model no merely he number of oble value aken by varable. n a VC model every effec even v k become rue only when uored by one or more caue even and here only a k where v k rue for any. An AVC model on he oher hand doe no requre any uor for v. v become rue (ha v become fale when v no uored by any caue even. For examle le x x 2 and y mean Weaher eabed Color-of-ea-urface reecvely. Ther oble value mh be {fne cloudy rany {and rock coralreef and {blue reen ray. f he color of ea urface deenden on he weaher and/or he eabed x x 2 and y comoe a VC model. A an examle of an AVC model uoe ha x mean deae and ha y are her ymom. Thee varable ake rue ( v or fale ( v a her value for he deae and he ymom. n h cae x and y comoe an AVC model becaue ymom do no aear whou any deae. 3. CONDTONAL CAUAL POBLTY n Defnon 2. menoned ha he value of an effec varable deenden on hoe of ome caue varable. The deendence however omeme unceran n he real world. n order o coe wh he uncerany we nroduce Pobly heory [67] no he caual model. 3. General Proere of Poble Le w and w 2 be varable akn a value n a e W and W 2 reecvely. Marnal obly drbuon on W and W 2 are exreed n (w and (w 2. Then he follown equaon hold n eneral [67]. 83
3 ( w ( w2 ³ ( w w2 (8 ( w w = ( w ( w w (9 2 2 where ( mean local roduc (local um when ued for even whle mean mn (max for obly drbuon. ( w w2 equvalen o on obly drbuon ( w w2. Poblc ndeendence and non-neracon are defned a follow: [Defnon 3.] w 2 oblcally ndeenden of w f and only f he nex equaon hold. ( w2 w = ( w2. (0 [Defnon 3.2] w and w 2 are non-neracve f and only f he nex equaon hold. ( w w2 = ( w ( w2. ( From eq. (9 and (0 w and w 2 are non-neracve f w 2 (w oblcally ndeenden of w (w Condonal Caual Pobly for a VC model Th econ defne Condonal Caual Pobly (CCPO for a VC model and examne he relaon beween convenonal condonal oble and CCPO. Noe ha x and y comoe a VC model hrouhou he econ. [Defnon 3.3] Le (x and (y be marnal obly drbuon on U and V reecvely. (y :x a obly drbuon of y :x on U V. Then ( y : x x called Condonal Caual Pobly drbuon for a VC model. [Defnon 3.4] n cae of a VC model x and y have caualy f and only f he nex equaon hold. $ ( hk ; ( vk : uh h > 0 uh ÎU vk ÎV. (2 [Defnon 3.5] n cae of a VC model a conuncon of cauaon and caue even a conex of y :x f nclude neher caue even of x nor cauaon even of y :x. [Defnon 3.6] Le X be a conex of y :x n a VC model. y :x oblcally cauaon ndeenden f and only f he nex equaon hold. ( y : x x X = ( y : x x. (3 Then we aume he follown: (a n cae of a VC model x (É oblcally ndeenden of each oher. (b n cae of a VC model y :x (É =ÉJ oblcally cauaon ndeenden. From he above aumon we can derve he follown rooon [5]. [Prooon 3.] n cae of a VC model y :x and x Õ ( ¹ are oblcally ndeenden of each oher. [Prooon 3.2] n cae of a VC model y :x and y Õ :x Õ ( ¹ are oblcally ndeenden of each oher. [Prooon 3.3] n cae of a VC model he nex equaon hold. ( y : x y : x Y = ( y : x Y ( y : x Y (4 where Y a conuncon of caue even and a neceary condon. ¹ no From he above he follown equaon are derved [5]. ( ( y = ( y : uh h ( uh (5 uh ÎU... ( y x... x = ( y : x x ( ( y : uh h ( uh (6 uh ÎU + ( y... yq x... x = ( y x... x (7 = q 84
4 where and q J. oblcally cauaon ndeenden f and only f he nex equaon hold. 3.3 Condonal Caual Pobly for an AVC model The econ defne CCPO for an AVC model and examne he relaon beween condonal oble and CCPO. Throuhou h econ x and y comoe an AVC model. [Defnon 3.7] Le obly drbuon on U = { u u V = { v v and Z = { v : u v : u be (x (y and (z reecvely. Then ( z x called Condonal Caual Pobly drbuon for an AVC model. Pleae noe ha CCPO drbuon for an AVC model defned on U Z no on U V lke a VC model. From eq. (6 n Defnon 2.3 we can e he follown: ( v : u = ( v : u = ( v : u = 00.. (8 When u «F ( u «T he follown are derved from eq. ( and (2. v : u «v : uu T «v : uu u «F. (9 v : u «v : u T «v : u u «u «T. (20 Therefore we e he follown CCPO. ( v : u = 00.. (2 ( v : u = 0.. (22 [Defnon 3.8] n cae of an AVC model x and y have caualy f and only f he nex equaon hold. ( v : u > 0. (23 ( z x X = ( z x. (24 Then we aume he follown: (a n cae of an AVC model x (É oblcally ndeenden of each oher. (b n cae of an AVC model z (É =ÉJ oblcally cauaon ndeenden. From he above aumon we can derve he follown rooon [4]. [Prooon 3.4] n cae of an AVC model z and x Õ ( ¹ are oblcally ndeenden of each oher. [Prooon 3.5] n cae of an AVC model z and z ÕÕ ( ¹ are oblcally ndeenden of each oher. [Prooon 3.6] n cae of an AVC model he nex equaon hold. ( z z Y = ( z Y ( z Y (25 where Y a conuncon of caue even and a neceary condon. ¹ no From he above he follown equaon are derved [4]. ( = ì ( v : u ( u f y v ì ü ( y = íí ( ( v : u x ( x ý îxî{ u u þ f y = v î (26 [Defnon 3.9] n cae of an AVC model a conuncon of cauaon even caue even and/or her neaon a conex of z f nclude none of u u v : u and v : u. [Defnon 3.0] Le X be a conex of z n an AVC model. z 85
5 ( y x... x ì ( v : u x ( ( v : u ( u f y = v + (27 = í ( v : u x ì ü í ( ( v : u x ( x ý + îxî{ u u þ î f y = v ( y... yq x... x = ( y x... x (28 = q where and q J. Po( P QT P Q º ( x y... u v... vq = + = q+ T x ÎP Í X - P T J y ÎQT ÍY -Q (29 where u... u v... v are he oberved value. q The above CCP reduced o an nvere roblem of caualy uded n [4] when P =Æ P = X and Q T =Æ. Caualy Analy Problem dcued n [5] a ecal cae where P =Æ and Q T = 0 or P =Æ and Q T =Æ. n h ene CCP an exended roblem o analyze a wo-layered herarchcal caual model. Un he roery of eq. (0 he nex equaon derved. 4. CAUALTY CONTENCY PROBLEM Th chaer defne a Caualy Conency Problem whch alcable boh o a VC and an AVC model and dcue he oluon. The dcuon can be conduced wh condonal oble derved from CCPO n he way hown n he revou chaer. n oher word we do no have o be cauou wh he dfference beween he wo model hrouhou he dcuon exce he roof of Lemma Problem Defnon and oluon Caualy Conency Problem defned a follow. [Defnon 4.] Le X be a e of caue varable x (É and Y be a e of effec varable y (=ÉJ. Marnal obly drbuon of he caue varable and CCPO drbuon are ven a a ror knowlede (Noe ha he defnon of CCPO are dfferen beween a VC and an AVC model. Then uoe ha value of x Î P ={ x... x and y Î Q ={ y... y q q J are oberved and ha hoe of he oher varable are unknown. Caualy Conency Problem (CCP one o oban a obly of combned value of ome unknown varable n P = { x+... x Í X -P and QT = { yq+... yt ÍY -Q under he ven condon. ( x y... u v... vq = + = q+ T ( v... q... ( v u u u... u (... q = v v y... u u = q+ T = + ( u... u u. = + (30 Le he 2nd and 3rd erm n he lef de of he equaon be E(PQ and he rh de be H(PQP Q T. Then hee are exreed a follow reardle of whch caual model baed on a VC or an AVC model houh he way o calculae he condonal oble are dfferen a dcued before. EPQ ( = ( v... u ( u. (3 = q HPQP ( QT = ( v... u x = q = + ( y... u x = q+ T = + ( u ( x. = + (32 E(PQ he obly ha u... u v... v are oberved. H(PQP Q T ve he obly ha he oberved value and value ecfed by x Î P and y Î QT haen mulaneouly. Un E(PQ and H(PQP Q T eq. (30 rewren a follow: q 86
6 PoP ( QT PQ EPQ ( = H( PQP QT. (33 Ulzn he dea of he lea ecfc oluon [8] whch chooe he oluon wh he reae obly deree n areemen wh he conran by eq. (33 he oluon of Caualy Conency Problem obaned n he nex equaon. Po( P QT P Q ìhpqp ( QT f EPQ ( > HPQP ( QT (34 = í î f EPQ ( = HPQP ( QT. 4.2 ome Proere of oluon ome roere of he oluon ven by eq. (34 are dcued. Thee roere hold n boh cae of a VC and an AVC model. [Prooon 4.] Po({ x Æ P Q Po( Æ{ y P Q xîp yîqt ³ Po( P QT P Q (35 Proof: The defnon ven by eq. (29 how ha eq. (35 hold f ( w w3 ( w2 w3 ³ ( w w2 w3. Aume ha ( w w3 ( w2 w3 < ( w w2 w3. Thu ( w w3 ( w2 w3 ( w3 < ( ww2 w3 ( w3 hold for w 3 afyn ( w3 > ( w w3 ( w3 > ( w2 w3 and ( w3 ³ ( w w2 w3. Thouh h lead o ( ww3 ( w2 w3 < ( ww2 w3 conradc eq. (8. Th becaue he aumon fale. (EOP [Prooon 4.2] Po( Æ { y P Q = Po( Æ QT P Q (36 yîqt Proof: Le QT y y Po( Æ{ y y P Q = {. Then he nex hold. ìhpq ( Æ { y y f EPQ ( > HPQ ( Æ{ y y = í î f EPQ ( = HPQ ( Æ{ y y. From eq. (32 we e H( PQ Æ { y y = H( PQ Æ{ y H( PQ Æ { y. Therefore he nex equaon hold. Po( Æ { y y P Q = Po( Æ{ y P Q Po( Æ { y P Q. Th clearly lead o eq. (36. (EOP [Lemma 4.] Le P X P Í - P X P 2 Í - Q X Q T Í - and Q X Q T 2 Í -. Then he follown hold. HPQP ( QT = HPQP ( QT 2 2 «Po( P QT P Q = Po( P QT P Q 2 2 HPQP ( QT > HPQP ( QT 2 2 «Po( P QT P Q > Po( P QT P Q 2 2 Proof: Thee are ealy roved from eq. (34. (EOP [Lemma 4.2] The nex equaon hold for x x ÎX - P. HPQ ( { x HPQ ( { x x. (37 (38 Æ ³ Æ (39 Proof: (a Cae of a VC model Le F({x x be defned a F({ x x = ( y : u { h h h uh ÎU ¹ = + ( y : u ( u. Thu he follown are derved from eq. (6. ( y... u x x ( x ( x = [ F({ x x ( y : x x ( y : x x ] ( x ( x éf({ x x { ( y : x x ( x ù = { ( y : x x ( x ( x ( x. ( y... u x ( x éf({ x x { ( y : x x ( x ù = ì ü ( x. í ( ( y : uh h ( uh ý î uh ÎU þ From he above he nex equaon obaned. ( y... u x ( x ³ ( y... u x x ( x ( x. Condern eq. (32 obvou ha eq. (39 hold n cae of a VC model. (b Cae of an AVC model (b- When y = v le G ({x x be defned a G ({ x x = ( v : u { ¹ = + ( v : u ( u. Thu he follown are derved from eq. (27. 87
7 ( v... u x x ( x ( x = G({ x x ( v : u x ( v : u x ( x ( x ég ({ x x { ( v : u x ( x ù = { ( v : u x ( x ( x ( x. ( v... u x ( x [ ] [ { ] = G ({ x x ( v : u x ( v : u ( u ( x ég ({ x x { ( v : u x ( x ù = ( x. { ( v : u ( u Condern x = u or u and ( v : u = 00. he nex equaon hold. ( v... u x ( x ³ ( v... u x x ( x ( x. (b-2 When y = v le G 2 ({x x be defned a G ({ x x = ( v : u 2 ¹ + ì í î xî{ u u ü ( v : u x ( x ý. þ ( Thu he follown are derved from eq. (27. ( v... u x x ( x ( x { = G ({ x x ( v : u x ( x 2 { ( v : u x ( x. ( v... u x ( x = G ({ x x ( v : u x ( x 2 { ì ü í ( ( v : u x ( x ý. îxî{ u u þ Therefore he nex equaon hold. ( v... u x ( x ³ ( v... u x x ( x ( x. From he reul of (b- and (b-2 eq. (39 alo hold n cae of an AVC model. (EOP [Prooon 4.3] Po({ x Æ P Q ³ Po( P Æ P Q. (40 xîp Proof: obvou from lemma 4. and 4.2. (EOP [Prooon 4.4] Po( P Æ P Q Po( Æ Q P Q ³ Po( P Q P Q. (4 T T Proof: Th roved f he follown wo nequale hold. HPQP ( Æ ³ HPQP ( Q. T HPQ ( Æ QT ³ HPQP ( QT. The former obvou from eq. (32. The laer roved n a mlar way o lemma 4.. (EOP 5. NUMERCAL EXAMPLE Le u verfy ome roere dcued n econ 4.2 un a numercal examle of a VC model hown n [5]. Le X = { x x2 x3 U = { a b c( = 23 Y = { y y2 y3 and V = { d e f( = 23. ( x and ( y : u are ven a follow: P ( a ( b ( c = [ ]= [ ] = [ ( a ( b ( c ]= [... ] = [ ( a ( b ( c ]= [... ] P P é( ad ( ae ( a f ù é P = ( bd ( be ( b f = ( cd ( ce ( c f é3( ad 3( ae 3( a f ù é P 3 = 3( bd 3( be 3( b f = ( cd 3( ce 3( c f é2( ad 2( ae 2( a f ù é P 2 = 2( bd 2( be 2( b f = ( cd 2( ce 2( c f é22( ad 22( ae 22( a f ù é P 22 = 22( bd 22( be 22( b f = ( cd 22( ce 22( c f é32( ad 32( ae 32( a f ù é P 32 = 32( bd 32( be 32( b f = ( cd 32( ce 32( c f é33( ad 33( ae 33( a f ù é P 33 = 33( bd 33( be 33( b f = ( cd 33( ce 33( c f where ( a = ( x = a and ( ad= ( y = d: x = a x = a. Then uoe ha P = { x2 = b and Q = { y = e y3 = d are oberved. n h cae Condonal oble of combned value of ome unknown varable are obaned a follow un eq. (34. épo({ x = a Æ P Q ù é04 Po({ x Æ P Q = Po({ x = b Æ P Q = 0. Po({ x = c Æ P Q
8 épo({ x3 = a Æ P Q ù é0 Po({ x3 Æ P Q = Po({ x3 = b Æ P Q = 04. Po({ x3 = c Æ P Q 03. épo( Æ { y2 = d P Q ù é03 Po( Æ { y P Q = Po( Æ { y = e P Q = Po( Æ { y = f P Q Po({ x x3 Æ P Q épo({ x = a x3 = a Æ P Q ù é04 Po({ x = a x3 = b Æ P Q 04. Po({ x = a x3 = c Æ P Q 03. Po({ x = b x3 = a Æ P Q 0. = Po({ x = b x3 = b Æ P Q = 04. Po({ x = b x3 =c Æ P Q* 00. Po({ x = c x3 = a Æ P Q 02. Po({ x = c x3 = b Æ P Q 02. Po({ x = c x = c Æ P Q* Po({ x = b x3{ y2 P Q épo({ x = b x3 = a{ y2 = d P Q ù é03 Po({ x = b x3 = a{ y2 = e P Q 0. Po({ x = b x3 = a{ y2 = f P Q** 03. Po({ x = b x3 = b{ y2 = d P Q 03. = Po ({ x = b x3 = b{ y2 = e P Q = 04. Po({ x = b x3 = b{ y2 = f P Q** 03. Po({ x = b x3 = c{ y2 = d P Q 00. Po({ x = b x3 = c{ y2 = e P Q 00. Po({ x = b x3 = c{ y2 = f P Q 00. Po({ x x3 Æ P Q n he above how ha Prooon 4.3 afed. Cae wh aerk * are hoe where equaly doe no hold. hold n he oher cae. From Po({ x x3{ y2 P Q we can ee ha Prooon 4.4 afed. Cae wh ** are hoe where equaly doe no hold. We can alo verfy ha rooon 4. afed n he above examle. 6. CONCLUON The aer ndcaed ha caual model are clafed no wo dfferen ye - ymmercally and aymmercally value caual model - deendn on he roere of caue and effec varable when hey are modeled un cauaon even and condonal caual oble. alo howed ha he relaon beween condonal caual oble and convenonal condonal oble are dfferen beween hee wo model. Then Caualy Conency Problem defned and oluon were hown. ome roere of he oluon were alo dcued. A for he roere all of hem dcued n he aer hold boh n a VC and an AVC model becaue hey can be dcued n he level of condonal oble derved from condonal caual oble exce for a roof of a roery. Caualy Conency Problem an exended one n he ene ha nclude nvere roblem of caualy [4] a well a caualy analy roblem [5]. Therefore he rooed aroach could be alcable o many analycal roblem ha nclude uncerany due o he comlexy of he real world. REFERENCE [] Y. Pen J. A. Rea: A Probablc Caual Model for Danoc Problem olvn - Par : neran ymbolc Caual nference wh Numercal Probablc nference EEE Tran. y. Man and Cyber. Vol. MC-7 No (987 [2] Y. Pen J. A. Rea: A Probablc Caual Model for Danoc Problem olvn - Par : Danoc raey EEE Tran. y. Man and Cyber. Vol. MC-7 No (987 [3] Y. Pen J. A. Rea: Abducve nference Model for Danoc Problem-olvn rner-verla (990 [4] K. Yamada: Condonal Caual Pobly and Alcaon o nvere Problem of Cauale J. Jaan ocey of Fuzzy Theory and yem Vol. No (999 (n Jaanee [5] K. Yamada: Caual Reaonn Un Condonal Caual Poble o Exre Uncerane for Caualy Proc. FA99 (o aear [6] L. A. Zadeh: Fuzzy e a a ba for a heory of obly Fuzzy e and yem Vol (978 [7] E. Hdal: Condonal Poble ndeendence and Non-neracon Fuzzy e and yem Vol (978 [8] D. Dubo H. Prade: Poblc nference under Marx Form n H. Prade and C.V. Neoa (ed. Fuzzy Loc n Knowlede Enneern Verla T T Rhenland Kšln (986 89
(,,, ) (,,, ). In addition, there are three other consumers, -2, -1, and 0. Consumer -2 has the utility function
MACROECONOMIC THEORY T J KEHOE ECON 87 SPRING 5 PROBLEM SET # Conder an overlappng generaon economy le ha n queon 5 on problem e n whch conumer lve for perod The uly funcon of he conumer born n perod,
More informationCooling of a hot metal forging. , dt dt
Tranen Conducon Uneady Analy - Lumped Thermal Capacy Model Performed when; Hea ranfer whn a yem produced a unform emperaure drbuon n he yem (mall emperaure graden). The emperaure change whn he yem condered
More informationEP2200 Queuing theory and teletraffic systems. 3rd lecture Markov chains Birth-death process - Poisson process. Viktoria Fodor KTH EES
EP Queung heory and eleraffc sysems 3rd lecure Marov chans Brh-deah rocess - Posson rocess Vora Fodor KTH EES Oulne for oday Marov rocesses Connuous-me Marov-chans Grah and marx reresenaon Transen and
More informationMechanics Physics 151
Mechancs Physcs 5 Lecure 9 Hamlonan Equaons of Moon (Chaper 8) Wha We Dd Las Tme Consruced Hamlonan formalsm H ( q, p, ) = q p L( q, q, ) H p = q H q = p H = L Equvalen o Lagrangan formalsm Smpler, bu
More informationMechanics Physics 151
Mechancs Physcs 5 Lecure 9 Hamlonan Equaons of Moon (Chaper 8) Wha We Dd Las Tme Consruced Hamlonan formalsm Hqp (,,) = qp Lqq (,,) H p = q H q = p H L = Equvalen o Lagrangan formalsm Smpler, bu wce as
More informationTesting of Markov Assumptions Based on the Dynamic Specification Test
Aca olechnca Hungarca Vol. 8 o. 3 0 Teng of Markov Aumon Baed on he Dnamc Secfcaon Te Jana Lenčuchová Dearmen of Mahemac Facul of Cvl Engneerng Slovak Unver of Technolog Bralava Radkého 83 68 Bralava Slovaka
More informationThruster Modulation for Unsymmetric Flexible Spacecraft with Consideration of Torque Arm Perturbation
hruer Modulaon for Unymmerc Flexble Sacecraf wh onderaon of orue rm Perurbaon a Shgemune anwak Shnchro chkawa a Yohak hkam b a Naonal Sace evelomen gency of Jaan 2-- Sengen ukuba-h barak b eo Unvery 3--
More informationPHYS 705: Classical Mechanics. Canonical Transformation
PHYS 705: Classcal Mechancs Canoncal Transformaon Canoncal Varables and Hamlonan Formalsm As we have seen, n he Hamlonan Formulaon of Mechancs,, are ndeenden varables n hase sace on eual foong The Hamlon
More informationH = d d q 1 d d q N d d p 1 d d p N exp
8333: Sacal Mechanc I roblem Se # 7 Soluon Fall 3 Canoncal Enemble Non-harmonc Ga: The Hamlonan for a ga of N non neracng parcle n a d dmenonal box ha he form H A p a The paron funcon gven by ZN T d d
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 31 Signal & Syem Prof. Mark Fowler Noe Se #27 C-T Syem: Laplace Tranform Power Tool for yem analyi Reading Aignmen: Secion 6.1 6.3 of Kamen and Heck 1/18 Coure Flow Diagram The arrow here how concepual
More informationA Demand System for Input Factors when there are Technological Changes in Production
A Demand Syem for Inpu Facor when here are Technologcal Change n Producon Movaon Due o (e.g.) echnologcal change here mgh no be a aonary relaonhp for he co hare of each npu facor. When emang demand yem
More informationLecture 18: The Laplace Transform (See Sections and 14.7 in Boas)
Lecure 8: The Lalace Transform (See Secons 88- and 47 n Boas) Recall ha our bg-cure goal s he analyss of he dfferenal equaon, ax bx cx F, where we emloy varous exansons for he drvng funcon F deendng on
More informationAdvanced time-series analysis (University of Lund, Economic History Department)
Advanced me-seres analss (Unvers of Lund, Economc Hsor Dearmen) 3 Jan-3 Februar and 6-3 March Lecure 4 Economerc echnues for saonar seres : Unvarae sochasc models wh Box- Jenns mehodolog, smle forecasng
More informationグラフィカルモデルによる推論 確率伝搬法 (2) Kenji Fukumizu The Institute of Statistical Mathematics 計算推論科学概論 II (2010 年度, 後期 )
グラフィカルモデルによる推論 確率伝搬法 Kenj Fukuzu he Insue of Sascal Maheacs 計算推論科学概論 II 年度 後期 Inference on Hdden Markov Model Inference on Hdden Markov Model Revew: HMM odel : hdden sae fne Inference Coue... for any Naïve
More informationMechanics Physics 151
Mechancs Physcs 5 Lecure 0 Canoncal Transformaons (Chaper 9) Wha We Dd Las Tme Hamlon s Prncple n he Hamlonan formalsm Dervaon was smple δi δ Addonal end-pon consrans pq H( q, p, ) d 0 δ q ( ) δq ( ) δ
More information( ) () we define the interaction representation by the unitary transformation () = ()
Hgher Order Perurbaon Theory Mchael Fowler 3/7/6 The neracon Represenaon Recall ha n he frs par of hs course sequence, we dscussed he chrödnger and Hesenberg represenaons of quanum mechancs here n he chrödnger
More informationMatrix reconstruction with the local max norm
Marx reconrucon wh he local max norm Rna oygel Deparmen of Sac Sanford Unvery rnafb@anfordedu Nahan Srebro Toyoa Technologcal Inue a Chcago na@cedu Rulan Salakhudnov Dep of Sac and Dep of Compuer Scence
More informationA. Inventory model. Why are we interested in it? What do we really study in such cases.
Some general yem model.. Inenory model. Why are we nereed n? Wha do we really udy n uch cae. General raegy of machng wo dmlar procee, ay, machng a fa proce wh a low one. We need an nenory or a buffer or
More informationUNIVERSITAT AUTÒNOMA DE BARCELONA MARCH 2017 EXAMINATION
INTERNATIONAL TRADE T. J. KEHOE UNIVERSITAT AUTÒNOMA DE BARCELONA MARCH 27 EXAMINATION Please answer wo of he hree quesons. You can consul class noes, workng papers, and arcles whle you are workng on he
More informationIn the complete model, these slopes are ANALYSIS OF VARIANCE FOR THE COMPLETE TWO-WAY MODEL. (! i+1 -! i ) + [(!") i+1,q - [(!
ANALYSIS OF VARIANCE FOR THE COMPLETE TWO-WAY MODEL The frs hng o es n wo-way ANOVA: Is here neracon? "No neracon" means: The man effecs model would f. Ths n urn means: In he neracon plo (wh A on he horzonal
More informationEE241 - Spring 2003 Advanced Digital Integrated Circuits
EE4 EE4 - rn 00 Advanced Dal Ineraed rcus Lecure 9 arry-lookahead Adders B. Nkolc, J. Rabaey arry-lookahead Adders Adder rees» Radx of a ree» Mnmum deh rees» arse rees Loc manulaons» onvenonal vs. Ln»
More informationAn Example file... log.txt
# ' ' Start of fie & %$ " 1 - : 5? ;., B - ( * * B - ( * * F I / 0. )- +, * ( ) 8 8 7 /. 6 )- +, 5 5 3 2( 7 7 +, 6 6 9( 3 5( ) 7-0 +, => - +< ( ) )- +, 7 / +, 5 9 (. 6 )- 0 * D>. C )- +, (A :, C 0 )- +,
More informationThe MacWilliams Identity of the Linear Codes over the Ring F p +uf p +vf p +uvf p
Reearch Joural of Aled Scece Eeer ad Techoloy (6): 28-282 22 ISSN: 2-6 Maxwell Scefc Orazao 22 Submed: March 26 22 Acceed: Arl 22 Publhed: Auu 5 22 The MacWllam Idey of he Lear ode over he R F +uf +vf
More informationDesign of Recursive Digital Filters IIR
Degn of Recurve Dgtal Flter IIR The outut from a recurve dgtal flter deend on one or more revou outut value, a well a on nut t nvolve feedbac. A recurve flter ha an nfnte mule reone (IIR). The mulve reone
More informationFI 3103 Quantum Physics
/9/4 FI 33 Quanum Physcs Aleander A. Iskandar Physcs of Magnesm and Phooncs Research Grou Insu Teknolog Bandung Basc Conces n Quanum Physcs Probably and Eecaon Value Hesenberg Uncerany Prncle Wave Funcon
More informationNONLOCAL BOUNDARY VALUE PROBLEM FOR SECOND ORDER ANTI-PERIODIC NONLINEAR IMPULSIVE q k INTEGRODIFFERENCE EQUATION
Euroean Journal of ahemac an Comuer Scence Vol No 7 ISSN 59-995 NONLOCAL BOUNDARY VALUE PROBLE FOR SECOND ORDER ANTI-PERIODIC NONLINEAR IPULSIVE - INTEGRODIFFERENCE EQUATION Hao Wang Yuhang Zhang ngyang
More informationu(t) Figure 1. Open loop control system
Open loop conrol v cloed loop feedbac conrol The nex wo figure preen he rucure of open loop and feedbac conrol yem Figure how an open loop conrol yem whoe funcion i o caue he oupu y o follow he reference
More informationNotes on the stability of dynamic systems and the use of Eigen Values.
Noes on he sabl of dnamc ssems and he use of Egen Values. Source: Macro II course noes, Dr. Davd Bessler s Tme Seres course noes, zarads (999) Ineremporal Macroeconomcs chaper 4 & Techncal ppend, and Hamlon
More informationAN ANALYTICAL METHOD OF SOLUTION FOR SYSTEMS OF BOOLEAN EQUATIONS
CHAPTER 5 AN ANALYTICAL METHOD OF SOLUTION FOR SYSTEMS OF BOOLEAN EQUATIONS 51 APPLICATIONS OF DE MORGAN S LAWS A we have een in Secion 44 of Chaer 4, any Boolean Equaion of ye (1), (2) or (3) could be
More informationGENERATING CERTAIN QUINTIC IRREDUCIBLE POLYNOMIALS OVER FINITE FIELDS. Youngwoo Ahn and Kitae Kim
Korean J. Mah. 19 (2011), No. 3, pp. 263 272 GENERATING CERTAIN QUINTIC IRREDUCIBLE POLYNOMIALS OVER FINITE FIELDS Youngwoo Ahn and Kae Km Absrac. In he paper [1], an explc correspondence beween ceran
More informationF-Tests and Analysis of Variance (ANOVA) in the Simple Linear Regression Model. 1. Introduction
ECOOMICS 35* -- OTE 9 ECO 35* -- OTE 9 F-Tess and Analyss of Varance (AOVA n he Smple Lnear Regresson Model Inroducon The smple lnear regresson model s gven by he followng populaon regresson equaon, or
More information! " # $! % & '! , ) ( + - (. ) ( ) * + / 0 1 2 3 0 / 4 5 / 6 0 ; 8 7 < = 7 > 8 7 8 9 : Œ Š ž P P h ˆ Š ˆ Œ ˆ Š ˆ Ž Ž Ý Ü Ý Ü Ý Ž Ý ê ç è ± ¹ ¼ ¹ ä ± ¹ w ç ¹ è ¼ è Œ ¹ ± ¹ è ¹ è ä ç w ¹ ã ¼ ¹ ä ¹ ¼ ¹ ±
More informationOpen Chemical Systems and Their Biological Function. Department of Applied Mathematics University of Washington
Oen Chemcal Syem and Ther Bologcal Funcon Hong Qan Dearmen of Aled Mahemac Unvery of Wahngon Dynamc and Thermodynamc of Sochac Nonlnear Meococ Syem Meococ decron of hycal and chemcal yem: Gbb 1870-1890
More informationELIMINATION OF DOMINATED STRATEGIES AND INESSENTIAL PLAYERS
OPERATIONS RESEARCH AND DECISIONS No. 1 215 DOI: 1.5277/ord1513 Mamoru KANEKO 1 Shuge LIU 1 ELIMINATION OF DOMINATED STRATEGIES AND INESSENTIAL PLAYERS We udy he proce, called he IEDI proce, of eraed elmnaon
More informationIntroduction to Hypothesis Testing
Noe for Seember, Iroducio o Hyohei Teig Scieific Mehod. Sae a reearch hyohei or oe a queio.. Gaher daa or evidece (obervaioal or eerimeal) o awer he queio. 3. Summarize daa ad e he hyohei. 4. Draw a cocluio.
More information[ ] 2. [ ]3 + (Δx i + Δx i 1 ) / 2. Δx i-1 Δx i Δx i+1. TPG4160 Reservoir Simulation 2018 Lecture note 3. page 1 of 5
TPG460 Reservor Smulaon 08 page of 5 DISCRETIZATIO OF THE FOW EQUATIOS As we already have seen, fne dfference appromaons of he paral dervaves appearng n he flow equaons may be obaned from Taylor seres
More informationDiscussion Session 2 Constant Acceleration/Relative Motion Week 03
PHYS 100 Dicuion Seion Conan Acceleraion/Relaive Moion Week 03 The Plan Today you will work wih your group explore he idea of reference frame (i.e. relaive moion) and moion wih conan acceleraion. You ll
More informationNotes on cointegration of real interest rates and real exchange rates. ρ (2)
Noe on coinegraion of real inere rae and real exchange rae Charle ngel, Univeriy of Wiconin Le me ar wih he obervaion ha while he lieraure (mo prominenly Meee and Rogoff (988) and dion and Paul (993))
More informationSolution in semi infinite diffusion couples (error function analysis)
Soluon n sem nfne dffuson couples (error funcon analyss) Le us consder now he sem nfne dffuson couple of wo blocks wh concenraon of and I means ha, n a A- bnary sysem, s bondng beween wo blocks made of
More informationRandomized Perfect Bipartite Matching
Inenive Algorihm Lecure 24 Randomized Perfec Biparie Maching Lecurer: Daniel A. Spielman April 9, 208 24. Inroducion We explain a randomized algorihm by Ahih Goel, Michael Kapralov and Sanjeev Khanna for
More informationCHAPTER 7: SECOND-ORDER CIRCUITS
EEE5: CI RCUI T THEORY CHAPTER 7: SECOND-ORDER CIRCUITS 7. Inroducion Thi chaper conider circui wih wo orage elemen. Known a econd-order circui becaue heir repone are decribed by differenial equaion ha
More informationLecture 3 Topic 2: Distributions, hypothesis testing, and sample size determination
Lecure 3 Topc : Drbuo, hypohe eg, ad ample ze deermao The Sude - drbuo Coder a repeaed drawg of ample of ze from a ormal drbuo of mea. For each ample, compue,,, ad aoher ac,, where: The ac he devao of
More informationL N O Q. l q l q. I. A General Case. l q RANDOM LAGRANGE MULTIPLIERS AND TRANSVERSALITY. Econ. 511b Spring 1998 C. Sims
Econ. 511b Sprng 1998 C. Sm RAD AGRAGE UPERS AD RASVERSAY agrange mulpler mehod are andard fare n elemenary calculu coure, and hey play a cenral role n economc applcaon of calculu becaue hey ofen urn ou
More informationAn Improved Anti-windup Control Using a PI Controller
05 Third nernaional Conference on Arificial nelligence, Modelling and Simulaion An mroved Ani-windu Conrol Uing a P Conroller Kyohei Saai Graduae School of Science and Technology Meiji univeriy Kanagawa,
More informationPHYSICS 151 Notes for Online Lecture #4
PHYSICS 5 Noe for Online Lecure #4 Acceleraion The ga pedal in a car i alo called an acceleraor becaue preing i allow you o change your elociy. Acceleraion i how fa he elociy change. So if you ar fro re
More informationOP = OO' + Ut + Vn + Wb. Material We Will Cover Today. Computer Vision Lecture 3. Multi-view Geometry I. Amnon Shashua
Comuer Vson 27 Lecure 3 Mul-vew Geomer I Amnon Shashua Maeral We Wll Cover oa he srucure of 3D->2D rojecon mar omograh Marces A rmer on rojecve geomer of he lane Eolar Geomer an Funamenal Mar ebrew Unvers
More informationOutline. GW approximation. Electrons in solids. The Green Function. Total energy---well solved Single particle excitation---under developing
Peenaon fo Theoecal Condened Mae Phyc n TU Beln Geen-Funcon and GW appoxmaon Xnzheng L Theoy Depamen FHI May.8h 2005 Elecon n old Oulne Toal enegy---well olved Sngle pacle excaon---unde developng The Geen
More informationRobust Controller Design Using Loop-Shaping and the Method of Inequalities
Robu Conroller Degn Ung H Loo-Shang and he Mehod of Inequale J F Whdborne, I Polehwae and D-W Gu Conrol Syem Reearch Dearmen of Engneerng Unvery of Leceer Leceer LE 7RH UK Ocober 99, reved July 99, Ocober
More informationSpecification -- Assumptions of the Simple Classical Linear Regression Model (CLRM) 1. Introduction
ECONOMICS 35* -- NOTE ECON 35* -- NOTE Specfcaton -- Aumpton of the Smple Clacal Lnear Regreon Model (CLRM). Introducton CLRM tand for the Clacal Lnear Regreon Model. The CLRM alo known a the tandard lnear
More informationOrdinary Differential Equations in Neuroscience with Matlab examples. Aim 1- Gain understanding of how to set up and solve ODE s
Ordnary Dfferenal Equaons n Neuroscence wh Malab eamples. Am - Gan undersandng of how o se up and solve ODE s Am Undersand how o se up an solve a smple eample of he Hebb rule n D Our goal a end of class
More informationA L A BA M A L A W R E V IE W
A L A BA M A L A W R E V IE W Volume 52 Fall 2000 Number 1 B E F O R E D I S A B I L I T Y C I V I L R I G HT S : C I V I L W A R P E N S I O N S A N D TH E P O L I T I C S O F D I S A B I L I T Y I N
More information8. Queueing systems lect08.ppt S Introduction to Teletraffic Theory - Fall
8. Queueg sysems lec8. S-38.45 - Iroduco o Teleraffc Theory - Fall 8. Queueg sysems Coes Refresher: Smle eleraffc model M/M/ server wag laces M/M/ servers wag laces 8. Queueg sysems Smle eleraffc model
More informationEpistemic Game Theory: Online Appendix
Epsemc Game Theory: Onlne Appendx Edde Dekel Lucano Pomao Marcano Snscalch July 18, 2014 Prelmnares Fx a fne ype srucure T I, S, T, β I and a probably µ S T. Le T µ I, S, T µ, βµ I be a ype srucure ha
More informationNPTEL Project. Econometric Modelling. Module23: Granger Causality Test. Lecture35: Granger Causality Test. Vinod Gupta School of Management
P age NPTEL Proec Economerc Modellng Vnod Gua School of Managemen Module23: Granger Causaly Tes Lecure35: Granger Causaly Tes Rudra P. Pradhan Vnod Gua School of Managemen Indan Insue of Technology Kharagur,
More informationESS 265 Spring Quarter 2005 Kinetic Simulations
SS 65 Spng Quae 5 Knec Sulaon Lecue une 9 5 An aple of an lecoagnec Pacle Code A an eaple of a knec ulaon we wll ue a one denonal elecoagnec ulaon code called KMPO deeloped b Yohhau Oua and Hoh Mauoo.
More informationFX-IR Hybrids Modeling
FX-IR Hybr Moeln Yauum Oajma Mubh UFJ Secure Dervave Reearch Dep. Reearch & Developmen Dvon Senor Manaer oajma-yauum@c.mu.jp Oaka Unvery Workhop December 5 h preenaon repreen he vew o he auhor an oe no
More informationt = s D Overview of Tests Two-Sample t-test: Independent Samples Independent Samples t-test Difference between Means in a Two-sample Experiment
Overview of Te Two-Sample -Te: Idepede Sample Chaper 4 z-te Oe Sample -Te Relaed Sample -Te Idepede Sample -Te Compare oe ample o a populaio Compare wo ample Differece bewee Mea i a Two-ample Experime
More informationP a g e 5 1 of R e p o r t P B 4 / 0 9
P a g e 5 1 of R e p o r t P B 4 / 0 9 J A R T a l s o c o n c l u d e d t h a t a l t h o u g h t h e i n t e n t o f N e l s o n s r e h a b i l i t a t i o n p l a n i s t o e n h a n c e c o n n e
More informationTHE PREDICTION OF COMPETITIVE ENVIRONMENT IN BUSINESS
THE PREICTION OF COMPETITIVE ENVIRONMENT IN BUSINESS INTROUCTION The wo dmensonal paral dfferenal equaons of second order can be used for he smulaon of compeve envronmen n busness The arcle presens he
More informationChapter Lagrangian Interpolation
Chaper 5.4 agrangan Inerpolaon Afer readng hs chaper you should be able o:. dere agrangan mehod of nerpolaon. sole problems usng agrangan mehod of nerpolaon and. use agrangan nerpolans o fnd deraes and
More information1 Motivation and Basic Definitions
CSCE : Deign and Analyi of Algorihm Noe on Max Flow Fall 20 (Baed on he preenaion in Chaper 26 of Inroducion o Algorihm, 3rd Ed. by Cormen, Leieron, Rive and Sein.) Moivaion and Baic Definiion Conider
More informationOnline Supplement for Dynamic Multi-Technology. Production-Inventory Problem with Emissions Trading
Onlne Supplemen for Dynamc Mul-Technology Producon-Invenory Problem wh Emssons Tradng by We Zhang Zhongsheng Hua Yu Xa and Baofeng Huo Proof of Lemma For any ( qr ) Θ s easy o verfy ha he lnear programmng
More informationAdditional File 1 - Detailed explanation of the expression level CPD
Addtonal Fle - Detaled explanaton of the expreon level CPD A mentoned n the man text, the man CPD for the uterng model cont of two ndvdual factor: P( level gen P( level gen P ( level gen 2 (.).. CPD factor
More informationA New Generalized Gronwall-Bellman Type Inequality
22 Inernaonal Conference on Image, Vson and Comung (ICIVC 22) IPCSIT vol. 5 (22) (22) IACSIT Press, Sngaore DOI:.7763/IPCSIT.22.V5.46 A New Generalzed Gronwall-Bellman Tye Ineualy Qnghua Feng School of
More information! -., THIS PAGE DECLASSIFIED IAW EQ t Fr ra _ ce, _., I B T 1CC33ti3HI QI L '14 D? 0. l d! .; ' D. o.. r l y. - - PR Pi B nt 8, HZ5 0 QL
H PAGE DECAFED AW E0 2958 UAF HORCA UD & D m \ Z c PREMNAR D FGHER BOMBER ARC o v N C o m p R C DECEMBER 956 PREPARED B HE UAF HORCA DVO N HRO UGH HE COOPERAON O F HE HORCA DVON HEADQUARER UAREUR DEPARMEN
More informationLinear Response Theory: The connection between QFT and experiments
Phys540.nb 39 3 Lnear Response Theory: The connecon beween QFT and expermens 3.1. Basc conceps and deas Q: ow do we measure he conducvy of a meal? A: we frs nroduce a weak elecrc feld E, and hen measure
More informationTSS = SST + SSE An orthogonal partition of the total SS
ANOVA: Topc 4. Orhogonal conrass [ST&D p. 183] H 0 : µ 1 = µ =... = µ H 1 : The mean of a leas one reamen group s dfferen To es hs hypohess, a basc ANOVA allocaes he varaon among reamen means (SST) equally
More informationELEC 201 Electric Circuit Analysis I Lecture 9(a) RLC Circuits: Introduction
//6 All le courey of Dr. Gregory J. Mazzaro EE Elecrc rcu Analy I ecure 9(a) rcu: Inroucon THE ITADE, THE MIITAY OEGE OF SOUTH AOINA 7 Moulre Sree, harleon, S 949 V Sere rcu: Analog Dcoery _ 5 Ω pf eq
More informationGraduate Macroeconomics 2 Problem set 5. - Solutions
Graduae Macroeconomcs 2 Problem se. - Soluons Queson 1 To answer hs queson we need he frms frs order condons and he equaon ha deermnes he number of frms n equlbrum. The frms frs order condons are: F K
More informationComparison of Differences between Power Means 1
In. Journal of Mah. Analyss, Vol. 7, 203, no., 5-55 Comparson of Dfferences beween Power Means Chang-An Tan, Guanghua Sh and Fe Zuo College of Mahemacs and Informaon Scence Henan Normal Unversy, 453007,
More informationLet s treat the problem of the response of a system to an applied external force. Again,
Page 33 QUANTUM LNEAR RESPONSE FUNCTON Le s rea he problem of he response of a sysem o an appled exernal force. Agan, H() H f () A H + V () Exernal agen acng on nernal varable Hamlonan for equlbrum sysem
More informationControl Systems. Mathematical Modeling of Control Systems.
Conrol Syem Mahemacal Modelng of Conrol Syem chbum@eoulech.ac.kr Oulne Mahemacal model and model ype. Tranfer funcon model Syem pole and zero Chbum Lee -Seoulech Conrol Syem Mahemacal Model Model are key
More informationTheoretical analysis on the transient characteristics of EDFA in optical. fiber communication
Journal of aled cence and engeerg novaon Vol. No. 04 ISSN (r: 33-906 ISSN (Onle): 33-9070 Theorecal analy on he ranen characerc of EDFA ocal fber communcaon Hao Zqang,a, L Hongzuo,b, Zhao Tg,c Changchun
More informationNECESSARY AND SUFFICIENT CONDITIONS FOR LATENT SEPARABILITY
NECESSARY AND SUFFICIENT CONDITIONS FOR LATENT SEPARABILITY Ian Crawford THE INSTITUTE FOR FISCAL STUDIES DEPARTMENT OF ECONOMICS, UCL cemmap working paper CWP02/04 Neceary and Sufficien Condiion for Laen
More informationSound Transmission Throough Lined, Composite Panel Structures: Transversely Isotropic Poro- Elastic Model
Prde nvery Prde e-pb Pblcaon of he Ray. Herrc aboraore School of Mechancal Engneerng 8-5 Sond Tranmon Throogh ned, Comoe Panel Srcre: Tranverely Ioroc Poro- Elac Model J Sar Bolon Prde nvery, bolon@rde.ed
More informationEE Control Systems LECTURE 2
Copyrigh F.L. Lewi 999 All righ reerved EE 434 - Conrol Syem LECTURE REVIEW OF LAPLACE TRANSFORM LAPLACE TRANSFORM The Laplace ranform i very ueful in analyi and deign for yem ha are linear and ime-invarian
More informationChapter 6: AC Circuits
Chaper 6: AC Crcus Chaper 6: Oulne Phasors and he AC Seady Sae AC Crcus A sable, lnear crcu operang n he seady sae wh snusodal excaon (.e., snusodal seady sae. Complee response forced response naural response.
More informationLecture 12: HEMT AC Properties
Lecure : HEMT A Proeres Quas-sac oeraon Transcaacances -araeers Non-quas ac effecs Parasc ressances / caacancs f f ax ean ue for aer 6: 7-86 95-407 {407-46 sk MEFET ars} 47-44. (.e. sk an MEFET ars brefl
More informationComb Filters. Comb Filters
The smple flers dscussed so far are characered eher by a sngle passband and/or a sngle sopband There are applcaons where flers wh mulple passbands and sopbands are requred Thecomb fler s an example of
More informationLaplace Transformation of Linear Time-Varying Systems
Laplace Tranformaon of Lnear Tme-Varyng Syem Shervn Erfan Reearch Cenre for Inegraed Mcroelecronc Elecrcal and Compuer Engneerng Deparmen Unvery of Wndor Wndor, Onaro N9B 3P4, Canada Aug. 4, 9 Oulne of
More informationCS286.2 Lecture 14: Quantum de Finetti Theorems II
CS286.2 Lecure 14: Quanum de Fne Theorems II Scrbe: Mara Okounkova 1 Saemen of he heorem Recall he las saemen of he quanum de Fne heorem from he prevous lecure. Theorem 1 Quanum de Fne). Le ρ Dens C 2
More informationMath 10B: Mock Mid II. April 13, 2016
Name: Soluions Mah 10B: Mock Mid II April 13, 016 1. ( poins) Sae, wih jusificaion, wheher he following saemens are rue or false. (a) If a 3 3 marix A saisfies A 3 A = 0, hen i canno be inverible. True.
More informationNormal Random Variable and its discriminant functions
Noral Rando Varable and s dscrnan funcons Oulne Noral Rando Varable Properes Dscrnan funcons Why Noral Rando Varables? Analycally racable Works well when observaon coes for a corruped snle prooype 3 The
More informationNATIONAL UNIVERSITY OF SINGAPORE PC5202 ADVANCED STATISTICAL MECHANICS. (Semester II: AY ) Time Allowed: 2 Hours
NATONAL UNVERSTY OF SNGAPORE PC5 ADVANCED STATSTCAL MECHANCS (Semeser : AY 1-13) Tme Allowed: Hours NSTRUCTONS TO CANDDATES 1. Ths examnaon paper conans 5 quesons and comprses 4 prned pages.. Answer all
More informationImprovements on Waring s Problem
Imrovement on Warng Problem L An-Png Bejng 85, PR Chna al@nacom Abtract By a new recurve algorthm for the auxlary equaton, n th aer, we wll gve ome mrovement for Warng roblem Keyword: Warng Problem, Hardy-Lttlewood
More informationSSRG International Journal of Thermal Engineering (SSRG-IJTE) Volume 4 Issue 1 January to April 2018
SSRG Inernaonal Journal of Thermal Engneerng (SSRG-IJTE) Volume 4 Iue 1 January o Aprl 18 Opmal Conrol for a Drbued Parameer Syem wh Tme-Delay, Non-Lnear Ung he Numercal Mehod. Applcaon o One- Sded Hea
More informationME 321 Kinematics and Dynamics of Machines S. Lambert Winter 2002
ME 31 Kiemaic ad Dyamic o Machie S. Lamber Wier 6.. Forced Vibraio wih Dampig Coider ow he cae o orced vibraio wih dampig. Recall ha he goverig diereial equaio i: m && c& k F() ad ha we will aume ha he
More informationFundamentals of PLLs (I)
Phae-Locked Loop Fundamenal of PLL (I) Chng-Yuan Yang Naonal Chung-Hng Unvery Deparmen of Elecrcal Engneerng Why phae-lock? - Jer Supreon - Frequency Synhe T T + 1 - Skew Reducon T + 2 T + 3 PLL fou =
More informationRICCI TYPE IDENTITIES FOR BASIC DIFFERENTIATION AND CURVATURE TENSORS IN OTSUKI SPACES 1
wnov Sad J. Math. wvol., No., 00, 7-87 7 RICCI TYPE IDENTITIES FOR BASIC DIFFERENTIATION AND CURVATURE TENSORS IN OTSUKI SPACES Svetlav M. Mnčć Abtract. In the Otuk ace ue made of two non-ymmetrc affne
More informationNew approach for numerical solution of Fredholm integral equations system of the second kind by using an expansion method
Ieraoal Reearch Joural o Appled ad Bac Scece Avalable ole a wwwrabcom ISSN 5-88X / Vol : 8- Scece xplorer Publcao New approach or umercal oluo o Fredholm eral equao yem o he ecod d by u a expao mehod Nare
More informationAppendix H: Rarefaction and extrapolation of Hill numbers for incidence data
Anne Chao Ncholas J Goell C seh lzabeh L ander K Ma Rober K Colwell and Aaron M llson 03 Rarefacon and erapolaon wh ll numbers: a framewor for samplng and esmaon n speces dversy sudes cology Monographs
More informationJ i-1 i. J i i+1. Numerical integration of the diffusion equation (I) Finite difference method. Spatial Discretization. Internal nodes.
umercal negraon of he dffuson equaon (I) Fne dfference mehod. Spaal screaon. Inernal nodes. R L V For hermal conducon le s dscree he spaal doman no small fne spans, =,,: Balance of parcles for an nernal
More informationModule 3: Element Properties Lecture 1: Natural Coordinates
Module 3: Element Propertes Lecture : Natural Coordnates Natural coordnate system s bascally a local coordnate system whch allows the specfcaton of a pont wthn the element by a set of dmensonless numbers
More informationMALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES. Institute for Mathematical Research, Universiti Putra Malaysia, UPM Serdang, Selangor, Malaysia
Malaysan Journal of Mahemacal Scences 9(2): 277-300 (2015) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal homeage: h://ensemumedumy/journal A Mehod for Deermnng -Adc Orders of Facorals 1* Rafka Zulkal,
More informationWrap up: Weighted, directed graph shortest path Minimum Spanning Tree. Feb 25, 2019 CSCI211 - Sprenkle
Objecive Wrap up: Weighed, direced graph hore pah Minimum Spanning Tree eb, 1 SI - Sprenkle 1 Review Wha are greedy algorihm? Wha i our emplae for olving hem? Review he la problem we were working on: Single-ource,
More informationFTCS Solution to the Heat Equation
FTCS Soluon o he Hea Equaon ME 448/548 Noes Gerald Reckenwald Porland Sae Unversy Deparmen of Mechancal Engneerng gerry@pdxedu ME 448/548: FTCS Soluon o he Hea Equaon Overvew Use he forward fne d erence
More informationTHE POLYNOMIAL TENSOR INTERPOLATION
Pease ce hs arce as: Grzegorz Berna, Ana Ceo, The oynoma ensor neroaon, Scenfc Research of he Insue of Mahemacs and Comuer Scence, 28, oume 7, Issue, ages 5-. The webse: h://www.amcm.cz./ Scenfc Research
More informationSample Final Exam (finals03) Covering Chapters 1-9 of Fundamentals of Signals & Systems
Sample Final Exam Covering Chaper 9 (final04) Sample Final Exam (final03) Covering Chaper 9 of Fundamenal of Signal & Syem Problem (0 mar) Conider he caual opamp circui iniially a re depiced below. I LI
More informationIntroduction to SLE Lecture Notes
Inroducion o SLE Lecure Noe May 13, 16 - The goal of hi ecion i o find a ufficien condiion of λ for he hull K o be generaed by a imple cure. I urn ou if λ 1 < 4 hen K i generaed by a imple curve. We will
More informationOutline. Energy-Efficient Target Coverage in Wireless Sensor Networks. Sensor Node. Introduction. Characteristics of WSN
Ener-Effcen Tare Coverae n Wreless Sensor Newors Presened b M Trà Tá -4-4 Inroducon Bacround Relaed Wor Our Proosal Oulne Maxmum Se Covers (MSC) Problem MSC Problem s NP-Comlee MSC Heursc Concluson Sensor
More information