Optimization under uncertainty: robust optimization
|
|
|
- Ανάργυρος Φραγκούδης
- 7 χρόνια πριν
- Προβολές:
Transcript
1 CES 735/8 Imre Pólik Outline Motivation Theoretical background Applications Optimization under uncertainty: robust optimization Imre Pólik, PhD McMaster University School of Computational Engineering and Science March 6, 2008
2 CES 735/8 Imre Pólik Outline Outline Motivation Theoretical background 1 Motivation The role of uncertainty Applications 2 Theoretical background Robust linear optimization Robust least squares optimization 3 Applications Truss topology design Antenna array design
3 Outline CES 735/8 Imre Pólik Motivation The role of uncertainty Theoretical background Applications Why uncertainty? Design on computer, measure with a ruler, cut with an axe! Uncertain data measurement errors noise unpredictability vibration Implementation errors technological limits imperfect materials
4 CES 735/8 Imre Pólik Robust linear optimization Outline Motivation Theoretical background Robust linear optimization Robust least squares optimization Applications
5 Robust linear programming the parameters in optimization problems are often uncertain, e.g., in an LP there can be uncertainty in c, a i, b i minimize c T x subject to a T i x b i, i = 1,..., m, two common approaches to handling uncertainty (in a i, for simplicity) deterministic model: constraints must hold for all a i E i minimize c T x subject to a T i x b i for all a i E i, i = 1,..., m, stochastic model: a i is random variable; constraints must hold with probability η minimize c T x subject to prob(a T i x b i) η, i = 1,..., m Convex optimization problems 4 26
6 deterministic approach via SOCP choose an ellipsoid as E i : E i = {ā i P i u u 2 1} (ā i R n, P i R n n ) center is ā i, semi-axes determined by singular values/vectors of P i robust LP minimize c T x subject to a T i x b i a i E i, i = 1,..., m is equivalent to the SOCP minimize c T x subject to ā T i x P i T x 2 b i, i = 1,..., m (follows from sup u 2 1(ā i P i u) T x = ā T i x P i T x 2) Convex optimization problems 4 27
7 stochastic approach via SOCP assume a i is Gaussian with mean ā i, covariance Σ i (a i N (ā i, Σ i )) a T i x is Gaussian r.v. with mean āt i x, variance xt Σ i x; hence ( ) prob(a T b i ā T i i x b i ) = Φ x Σ 1/2 i x 2 where Φ(x) = (1/ 2π) x e t2 /2 dt is CDF of N (0, 1) robust LP minimize c T x subject to prob(a T i x b i) η, i = 1,..., m, with η 1/2, is equivalent to the SOCP minimize c T x subject to ā T i x Φ 1 (η) Σ 1/2 i x 2 b i, i = 1,..., m Convex optimization problems 4 28
8 CES 735/8 Imre Pólik Robust least squares optimization Outline Motivation Theoretical background Robust linear optimization Robust least squares optimization Applications
9 minimize Ax b with uncertain A Robust approximation two approaches: PSfrag replacements stochastic: assume A is random, minimize E Ax b worst-case: set A of possible values of A, minimize sup A A Ax b tractable only in special cases (certain norms, distributions, sets A) example: A(u) = A 0 ua 1 x nom minimizes A 0 x b x nom x stoch minimizes E A(u)x b 2 2 with u uniform on [ 1, 1] r(u) 6 4 x stoch x wc x wc minimizes sup 1 u 1 A(u)x b figure shows r(u) = A(u)x b u Approximation and fitting 6 17
10 stochastic robust LS with A = Ā U, U random, E U = 0, E U T U = P minimize E (Ā U)x b 2 2 explicit expression for objective: E Ax b 2 2 = E Āx b Ux 2 2 = Āx b 2 2 E x T U T Ux = Āx b 2 2 x T P x hence, robust LS problem is equivalent to LS problem minimize Āx b 2 2 P 1/2 x 2 2 for P = δi, get Tikhonov regularized problem minimize Āx b 2 2 δ x 2 2 Approximation and fitting 6 18
11 worst-case robust LS with A = {Ā u 1A 1 u p A p u 2 1} minimize sup A A Ax b 2 2 = sup u 2 1 P (x)u q(x) 2 2 where P (x) = [ A 1 x A 2 x A p x ], q(x) = Āx b from page 5 14, strong duality holds between the following problems maximize P u q 2 2 subject to u minimize subject to t λ I P q P T λi 0 q T 0 t 0 hence, robust LS problem is equivalent to SDP minimize subject to t λ I P (x) q(x) P (x) T λi 0 q(x) T 0 t 0 Approximation and fitting 6 19
12 PSfrag replacements example: histogram of residuals r(u) = (A 0 u 1 A 1 u 2 A 2 )x b 2 with u uniformly distributed on unit disk, for three values of x x rls frequency x tik x ls r(u) x ls minimizes A 0 x b 2 x tik minimizes A 0 x b 2 2 x 2 2 (Tikhonov solution) x wc minimizes sup u 2 1 A 0 x b 2 2 x 2 2 Approximation and fitting 6 20
13 CES 735/8 Imre Pólik Truss topology design Outline Motivation Theoretical background Applications Truss topology design Antenna array design
14 ? * } ('!4 1 1 S $ 1 1 /P $ / / 7P / ² Ÿ $ ² 1 QP R P / P 1 P $O S S P --! " $# *) * -" *),$# $ "! ' $ ( (!4 $ $# $H ' $(! $# $$!( " % $$$! $# ' 5 (! ( $(! $# $,$$! š $ "! "! 5 ( $ $
15 !( )!" $ 5 "0$($! "! * " / š 0 $ '"! # ) $ 0 $ '"!)* ) % ($ $# $# ) % ( $ - $! # (!( " $* "! /)& $#!( $" " ) ' "!4 - $#! A B h: db > h y $# - $!( $" " $#!( " )! ' "- )! ' $# ($ $# * & $ š - $!( $" " $# )!! "! $ ' $ )! # $ "- $ $ ' $(! # )! 5 "! ) ( ',., 2 D " $ " $* 5 œ $# " - ( 5 -,$# ( -" )* $ * $# )!$ $# " $* $!( /!( $" " )*! $% $'! "! ' / $' (0 &!( $" " $# )!
16 ± 5 % ($,$# - $%!( $" " %,$#( " $ " $* 5 % $# " $* $!(!( $'& " ) ) $ $$! ("! $ " "! )* -" $# $$!"! )* -" )* $ 4 (' 5 #! $( 5 4 ( -$ 1$# "- 25 ³ / $( ' $(! $$! 5 % ($,$# - $%!( $" " %,$#( " $ " $* 5 % 5 $# " $* $!(!( $'& " $($# $$! ³ ) ) )! "(! "0$($ 0 $ $# "" " š & ' $(! 0 $!,! )"' 5 $* ',-",.! $% $# $ "!)*"'& " $
17 % % H A G $ W N ' F % ) 5 % ($,$# - $%!( $" " %,$#( " $ " $* 5 % ) ) 5 $# " $* $!(!( $'& " $($# $$! ³ ) ) ). - 5 ' $ $ 5 -$! # ( - ' '! -$!,# 5 ( "" " & ( $#1 " Y a ( d 4 Y 8>`*f]d [a!za@b5y %l^bf]g^5fb5y a!di'z ]b5ea- ghy[z e*\a-a!\ Y[d ;e*e d 8>`a a!g`y[z]ighyo\y!^y [b5ef]z]\ ^bf]g^5fb5a-, b5a " 8>`acd]Y[gha#> \ e;eoib^5fy \]idy[gha- a!z^5dck " 2 df=;da.^ H I > e;0$iza- Y!^5ighY*' Y[\4 iddi'a7\]id 9 Y[gha- a!z^5dck= a-e K " 2 ghe*a!g^5iez Q FG mhmhm G L FG mhmhm G Y!^bigha!d,\a.- Z]iZ=1^5`aA(a!Za@bM Y!^bï 54, N Q S R W QX X Q S Q Q G S SPQ R N % QX a!iz=1^5`a1\a!di'zmo Y biy a!d Z 6 R > Y I Q S R = K 6 Q S R T X G D Q S Q R m " 2 ^ 4 Y d 4 i g 4 Y - Y i ^ R Q S R = Q S "2 a:5*^a@bzy1e[y[\ id b5a6]b5a!da!z^a!\)<7ymoa!g^e[b ^5`a1gheiY[Z]gha e; ghez]d ^bf]g^5iez Sb ^,6 id 9;:5<>=@?A B2C G `]i*a ^5`a Y!^a@biY b5a!defbghacid QX <;ighy^5i* Y %l^bf]g^5fb5y a!di'zm]b5ea- jlk idc i'oa!z 2Y [b5ef]z]\ ^bf]g^5fb5ap Y[Z7f=4 a@b ef]z]\ ez ^e ^Y1 Y!^a@biY8b5a!defbgha YeZ= ^5` ef]z]\]d ez&^5`a1b5a!defbgha!d e;ua-a- a!z^5d;y[z]\ da.^! e;e[y[\]iz=7dgha!zy bied Z]\ ghez]d ^bf]g^5iez ^5`1^5`a dv Y*a!d eddi'a">e[bd ^ghy[da# Sb ^ >6!;gheiY[Z]gha 9;:5<>=@?%$ B2C 9;:5<>=@?A R& <('*). Q S Q,- G Q S Q R. Q G FG mhmhm G L G N jlk W Q S Q R QX jlk
18 N A G $ W Q " 2 ^ Q S R = Q S <;ighy %l^bf]g^5fb5y a!di'zm]b5ea- id 9;:5<>=@?%$ B2C 9;:5<>=@?A R& <('*). Q S Q, - G Q S Q R. Q G FG mhmhm G L G N jlk W Q S Q R QX jlk `a@b5a " 9;:5<>=@? A Q S R B2C = D G 6 Q S R " Q S R QX X Q S Q Q "3H R \ "S Q R % FG mhmhm G L*Y b5a1\a!di'zmo Y biy a!d " I \ id>^5`a1da.^ e;ee[y[\]iz= dgha!zy bied "&.b5e Ze ez >acy[ddf4 a ^5`Y!^ 2 4 H dy!^5id -;a!d^5`a %=Y!^a@b ghez]\]i ^5ieZ7! f4 ^5ï 9 e[y[\y[z]\ e;f]d ^&%l^bf]g^5fb5y a!di'z " Z ^5`a f]dfyda.^^5iz=d e;7^5`a % ]b5ea- 6 G mhmhm G 6 idy dv Y*O- Z]i ^a da.^e;a(ce[y[\]d_e; iz^a@b5a!d ^, (C1f4 ^5ï 9 e[y[\&\a!di'z4, 4 2 Z Y ^a@bzy!^5i'oaid ^e f]da Y (C Y[ddi'Oa-,&a-*i';dei\Y da.^ e;e[y[\]dc 6 F R#! \ " 8>`a 1f4 ^5ï 9 e[y[\7y 4]b5e[Y[g`7Y[g`]ia6Oa!d,^5`a a!d ^ eddi'a b5a!df4 ^5dlY[d ;Y b>y[d ^5`a dgha!zy bie#e[y[\]d Y b5a ghez]gha@bza!\ ;f*^ ghy[zjia-\_f]z]d ^Y acghez]d ^bf]g^5iez `]ig`&ghy[zj a1ghbf]d`a!\ < (hdv Y* e*g.ghy[diezy e[y[\4, d ^ i ^ e W - Q 4 -. Q. Q. Q,4 8>`aA[b5ef]Z]\ ^bf]g^5fb5a \ela!d Ze Y[\4 bi'i\ e*\= ^5ieZ]dC Q Q Y[Z]\
19 /E5Ya ddf4 a!>acy b5a1\a!di'z]iz=1yy[zy ^bf]dd _Y ^bf]g^5fb5a Y[Z]\^5`a ez4' e[y[\ ^O6 b5acy[d d`e Z&eZ_^5`a#;ig^5fb5a [b5ef]z]\d ^ 8>`a_e^5i* Y>diZ=a:9 e[y[\ \a!di'z ia-\]d Y Z]igha^bf]ddcY[d ;e*e dc ^5i* Y ghy[z^5i*a6oa@b diz=a:9 e[y[\_\a!di'z 4 ^5`a1gheiY[Z]ghacid ^5i* Y ghy[z^5i*a6oa@b diz=a:9 e[y[\_\a!di'z 4 ^5`a1gheiY[Z]ghacid "^c^5fbz]d ef*^1^5`y!^ `a!z ^5`a (Ce[Y[\ e; iz^a@b5a!d ^,76 id b5a6y[gha!\ i ^5` ^5`a7dV Y*16 - m ( ]Y[\4' Y[gha!\4,1e*g.ghY[dieZY7e[Y[\&6 l^5`a gheiy[z]gha5f4;d ;b5e ^e 4 " Z e[b\a@b ^eci*]b5e-oa,^5`aod ^Y ;i*i ^ ce;^5`a \a!di'z a.^ f]d b5a6y[gha ^5`a&diZ=a e[y[\ e;,iz^a@b5a!d ^ 6<7^5`aa-*i';dei\ ghez^y[iz]iz= ^5`]idUe[Y[\Y[Z]\Y* e[y[\]d61e; Y Z]i ^5f]\a86 Ze ^ a:5]gha@a!\]iz= e;8^5`a Y Z]i ^5f]\ace; mhmhm! 6! F G `a@b5a,6 G mhmhm G 6! Y b5a>e;^5`aze[b - m F 86" Y[Z]\16 G 6 G mhmhm G 6! idy[z&e[b^5`e[ezy ]Y[did,iZM>!
20 " Y[ddiZ=&;b5e ^5`aOdiZ=a:9 e[y[\ ^e ^5`a b5e;f]d ^ \a!di'z >a e*\]i'; ^5`acb5a!df4 ^,Y[d ;e*e dc ^5i* Y ghy[z^5i*a6oa@b diz=a:9 e[y[\_\a!di'z 4 ( e;f]d ^, ghy[z^5i*a6oa@b eiy[z]gha!d a!di'z %iz=a:9 e[y[\ e;f]d ^ eiy[z]ghä Sb ^,6 < gheiy[z]ghä Sb ^ e[y[\]d m F 86"
21 CES 735/8 Imre Pólik Antenna array design Outline Motivation Theoretical background Applications Truss topology design Antenna array design
22 I I ˆ D LP R P 1 P $ R 1-5 P P5 5 ŸP 1 6 F I -[ 6 F I -[ F F I 1. % Œ I,1F -[ F F I 1 % Œ I,1F -[!! Œ! DO / 1L: 6 F I -[ A- N : % Œ 1F -[ F I6F IA- :F IoD> 1 F I P7 žp R 3 F I 6 F I 3. «P 1 2P R P D % - P % 0. P R Ia9 * } D! * $O -ŸP F$#*I / «1 P /.š % «P "AF$#*Ia9.>?>?>?9 F$#*I % 5 QP 1 $.š 0 C J. 1 1 C ˆ) 0 C C F$#*I - $7 -²P F$#*IaR R 4 $,' Q P C F$#*I - $ 0 C Ÿ QP / QP 1P 1QP / /š P P= 5 QP P D P DO R / 5 P P 1 "' D P - 1 D. DO - AO P P QP P $ O! F I % C ˆ) 0 C C F I./ 1 D. DO 4 R
23 y * k! N - ² P - QP / P «.. š! * P / AO P QP5P QP P 1 QP Q$4! y F I6 ƒ < 3 F Œ [ F IF /I!! P S / 1 QP J 8-9 ' Œ 6 R
24 ! % 3 *. 5 -LP AO P 1 QP PQš 7 / 1.2 *.Q / P - - R! ²P P -$# QP (' ž 4 O QP! C F IQ6 % F I %& / 3 / C )-,DC C F I E J F mf IkF /I!"! C 6 8->?<ZG94GŸ6«8-9.<9.>?>?>?9k<8 «6 3 ) ²P P. P QP k P P P / $ QP P P $!!!!! "!
25 e } } N! DO P Q L - 1! * $O QP -žp! % F I % C ),DC C F I 9 J S C F$#*I - F$#*I 0 ² / ² - P LP! F I % C ),DC C F I! ž P 8-9 % F I % C ),DC C F I 9 J P. P QP k D# P P P F I 6=<B& 6T8->8 / ; $ž, C ' G 6 <9.>?>?>?9.<8 ' 1 1 QP DO R ),' D P QPT P«O R Ÿ $ 1 P 0 R QP.!,DC 6=FZ< 1 ªHC I, C 1ªHC Z % 8->88-<9m8->88-<$ C )
26 I } $.Q R.š žp! S QP /š 1 šp 1 P P. P"! -²P ² -žp Z,' QP 5 Z P P 8->8 -R $# 1 P š. T 1 Ÿ L Z P R - P } H % F I % C ),DC C F I 9 J, š,DC FZ< 1 ªHC I,DC (' 7.. QP «7 Q ² žp P 8-> -9.<>88-< šq! Ž až 6 6 FZ< 1 ª m9.>?>?>?9.< 1 ª ªHC 8->88-<. O N ²P $ '/ QP T. 4 4 '. R DO 'N P 1$ Ž až QP O$O š % QP $ 7 1 QP QP QP QP ' 5 Z Q.$ 5 DO š«o D $ $$ R - P 5 ž D R
27 8 3 ' ' ' C ' ' 1 C } P 5 ) CA,DC QP -,DC QP $. P"! 8,DC FZ< 1 ªHC I,DC ª C / QP QP P QP O! 1ª C 6T8-B 1ª C 3 6 C 3 > 7O Qš QP $. P, C '; T L QP P QP O! 6 &/' C ) C FZ< 1 ª C I, C 1 6 &/' C ) CA,DC 1 9 F % I & C ' ) C 3 C 3, C 3 > -² š QP $ DO 2 Ÿ % D9 / Q $ Z / DO R 5 š 1 ' š O 8 % R R ' % ' C ) CA,DC 1 % ' C ) C 3 C 3, C3 R R R } &/' C ) C FZ< 1 ªHC I,DC 1 8 1ªHC 6«8-B 1ªa3 C 6 3 & C ' ) CA,DC 1 % & C ' ) C 3 C 3, C3 8! - 1$. P«T P Fm9.>?>?>?9 I < (' $ / DO.š $O / 1 - O. P - O 1 6 FZFZ< 1 ª ai m9.>?>?>?9.fz< 1 ª I I P R. 7ªHC 1 T / % ª 9 ª R ;0 ² P/ ª C QP5 T šo 1 - R )m $ % - A- % 3 DO š $ / Ÿ / <8 [ R ' P 3 o D $ $.š o
28 & } - $= P - P = -! % F I % C ), C C F I E J 4 }.Q LS $ Qš & 6 < Z! 1.2 S QP F I %& F I % C )-,DC C F I21 C )-,DC C F I % ª & C ), C 3 C3 F I ª & C ), C3 C3 F I % E J _ª 6T8->88-< š P TP. 1 Z P R - 0 / O š R $O ŸP $ Qš Ÿ 1 $ 5 T ' P $ $! $# 5P.š 'N Z P 7 -žo R R R
29 e c 3 & } P R ' P $ Qš 1$ P R.! $# P.š ' ² Z P Ÿ ž - O R R R } F I % F I %& % F I % C ),DC C F I "E J 4 C )-,DC C F I21 ª & C ), C 3 C3 F I C ), C C F I %ª & C ), C3 C3 F I % E J P $ Qš ² žp C F$#*I T 1$ 1$P P š R.Q ' ² PQš % / 7P P 1$ / š R )QP P ' Ÿ T / 4 PQP P, C T O! - o 1$ o o H o H o o o o / š / 4 2O ' QP DO $O - D P = TP $ - $ R ). ' QP QP P $#P '.² P - - / QP= QP R R 'Q R C ü *s e c H
30 / 3? P 2 ž 5 LP 1,'! * $O QP -žp!? S C F$#*I F$#*I QPT $2,DC ' GŸ6=<9.>?>?>?9.<8 $# / $O / F I % C ),DC C F I F I 3! 3! ² P $ $O # P -! - $-! LP $ R
31 ˆ I? "! š $ $O 7 NP $ 1 PL R QP. k D! P R O ' QP $# P ' 1 Z! ? N0. -Q 7 ",ž% 3 F 1 <ki, - ' 6=<8 R 1.š, ( FZ< 1 ª ( I, ( 9 ª ( J % ª 9 ª - 1$ Ÿ '"² 7. QP«O / L FZ< 1 ª m9.>?>?>?9.< 1 ª ª ( ª D9 / š R P $ -!N - $-! P.š R
32 ˆ I ˆ I? ",ž% 3 J 6 6 FZ< 1 ª m9.>?>?>?9.< 1 ª ª ( ª - P. - P 6 6 FZ< 1 ª 9.>?>?>?9.< 1 ª (*) ˆ ª 3( ª 3 - / 1; ; QP QP ' QP 5 QP. 1š D P 5 5. P QP $O.Q / ²! ",ž% 3 E J >.Q ' O P $= LP $ 'Q / / / 4! P $ -!N - $-! P.š R 270 $O 4 L R R
! " # $ % & $ % & $ & # " ' $ ( $ ) * ) * +, -. / # $ $ ( $ " $ $ $ % $ $ ' ƒ " " ' %. " 0 1 2 3 4 5 6 7 8 9 : ; ; < = : ; > : 0? @ 8? 4 A 1 4 B 3 C 8? D C B? E F 4 5 8 3 G @ H I@ A 1 4 D G 8 5 1 @ J C
Statistics 104: Quantitative Methods for Economics Formula and Theorem Review
Harvard College Statistics 104: Quantitative Methods for Economics Formula and Theorem Review Tommy MacWilliam, 13 tmacwilliam@college.harvard.edu March 10, 2011 Contents 1 Introduction to Data 5 1.1 Sample
Z L L L N b d g 5 * " # $ % $ ' $ % % % ) * + *, - %. / / + 3 / / / / + * 4 / / 1 " 5 % / 6, 7 # * $ 8 2. / / % 1 9 ; < ; = ; ; >? 8 3 " #
Z L L L N b d g 5 * " # $ % $ ' $ % % % ) * + *, - %. / 0 1 2 / + 3 / / 1 2 3 / / + * 4 / / 1 " 5 % / 6, 7 # * $ 8 2. / / % 1 9 ; < ; = ; ; >? 8 3 " # $ % $ ' $ % ) * % @ + * 1 A B C D E D F 9 O O D H
Other Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
) * +, -. + / - 0 1 2 3 4 5 6 7 8 9 6 : ; < 8 = 8 9 >? @ A 4 5 6 7 8 9 6 ; = B? @ : C B B D 9 E : F 9 C 6 < G 8 B A F A > < C 6 < B H 8 9 I 8 9 E ) * +, -. + / J - 0 1 2 3 J K 3 L M N L O / 1 L 3 O 2,
Fractional Colorings and Zykov Products of graphs
Fractional Colorings and Zykov Products of graphs Who? Nichole Schimanski When? July 27, 2011 Graphs A graph, G, consists of a vertex set, V (G), and an edge set, E(G). V (G) is any finite set E(G) is
Estimation for ARMA Processes with Stable Noise. Matt Calder & Richard A. Davis Colorado State University
Estimation for ARMA Processes with Stable Noise Matt Calder & Richard A. Davis Colorado State University rdavis@stat.colostate.edu 1 ARMA processes with stable noise Review of M-estimation Examples of
Bayesian statistics. DS GA 1002 Probability and Statistics for Data Science.
Bayesian statistics DS GA 1002 Probability and Statistics for Data Science http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall17 Carlos Fernandez-Granda Frequentist vs Bayesian statistics In frequentist
5. Choice under Uncertainty
5. Choice under Uncertainty Daisuke Oyama Microeconomics I May 23, 2018 Formulations von Neumann-Morgenstern (1944/1947) X: Set of prizes Π: Set of probability distributions on X : Preference relation
Ó³ Ÿ , º 2(214).. 171Ä176. Š Œ œ ƒˆˆ ˆ ˆŠ
Ó³ Ÿ. 218.. 15, º 2(214).. 171Ä176 Š Œ œ ƒˆˆ ˆ ˆŠ ˆ ˆ ˆ Š Š Œ Œ Ÿ ˆ Š ˆ Š ˆ ˆŠ Œ œ ˆ.. Š Ö,, 1,.. ˆ μ,,.. μ³ μ,.. ÉÓÖ μ,,.š. ʳÖ,, Í μ ²Ó Ò ² μ É ²Ó ± Ö Ò Ê É É Œˆ ˆ, Œμ ± Ñ Ò É ÉÊÉ Ö ÒÌ ² μ, Ê μ ± Ê É
Solutions to Exercise Sheet 5
Solutions to Eercise Sheet 5 jacques@ucsd.edu. Let X and Y be random variables with joint pdf f(, y) = 3y( + y) where and y. Determine each of the following probabilities. Solutions. a. P (X ). b. P (X
New bounds for spherical two-distance sets and equiangular lines
New bounds for spherical two-distance sets and equiangular lines Michigan State University Oct 8-31, 016 Anhui University Definition If X = {x 1, x,, x N } S n 1 (unit sphere in R n ) and x i, x j = a
Solution Series 9. i=1 x i and i=1 x i.
Lecturer: Prof. Dr. Mete SONER Coordinator: Yilin WANG Solution Series 9 Q1. Let α, β >, the p.d.f. of a beta distribution with parameters α and β is { Γ(α+β) Γ(α)Γ(β) f(x α, β) xα 1 (1 x) β 1 for < x
Probability and Random Processes (Part II)
Probability and Random Processes (Part II) 1. If the variance σ x of d(n) = x(n) x(n 1) is one-tenth the variance σ x of a stationary zero-mean discrete-time signal x(n), then the normalized autocorrelation
Statistical Inference I Locally most powerful tests
Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided
Probabilistic Approach to Robust Optimization
Probabilistic Approach to Robust Optimization Akiko Takeda Department of Mathematical & Computing Sciences Graduate School of Information Science and Engineering Tokyo Institute of Technology Tokyo 52-8552,
k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +
[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +](/thumbs/73/69566903.jpg "k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +")
Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b
Approximation of distance between locations on earth given by latitude and longitude
Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth
Numerical Analysis FMN011
Numerical Analysis FMN011 Carmen Arévalo Lund University carmen@maths.lth.se Lecture 12 Periodic data A function g has period P if g(x + P ) = g(x) Model: Trigonometric polynomial of order M T M (x) =
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question 1 a) The cumulative distribution function of T conditional on N n is Pr T t N n) Pr max X 1,..., X N ) t N n) Pr max
2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
Μηχανική Μάθηση Hypothesis Testing
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Μηχανική Μάθηση Hypothesis Testing Γιώργος Μπορμπουδάκης Τμήμα Επιστήμης Υπολογιστών Procedure 1. Form the null (H 0 ) and alternative (H 1 ) hypothesis 2. Consider
Matrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
5.4 The Poisson Distribution.
The worst thing you can do about a situation is nothing. Sr. O Shea Jackson 5.4 The Poisson Distribution. Description of the Poisson Distribution Discrete probability distribution. The random variable
ST5224: Advanced Statistical Theory II
ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known
Partial Differential Equations in Biology The boundary element method. March 26, 2013
The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet
Homework 8 Model Solution Section
MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx
4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
The ε-pseudospectrum of a Matrix
The ε-pseudospectrum of a Matrix Feb 16, 2015 () The ε-pseudospectrum of a Matrix Feb 16, 2015 1 / 18 1 Preliminaries 2 Definitions 3 Basic Properties 4 Computation of Pseudospectrum of 2 2 5 Problems
Lecture 34 Bootstrap confidence intervals
Lecture 34 Bootstrap confidence intervals Confidence Intervals θ: an unknown parameter of interest We want to find limits θ and θ such that Gt = P nˆθ θ t If G 1 1 α is known, then P θ θ = P θ θ = 1 α
LAD Estimation for Time Series Models With Finite and Infinite Variance
LAD Estimatio for Time Series Moels With Fiite a Ifiite Variace Richar A. Davis Colorao State Uiversity William Dusmuir Uiversity of New South Wales 1 LAD Estimatio for ARMA Moels fiite variace ifiite
An Inventory of Continuous Distributions
Appendi A An Inventory of Continuous Distributions A.1 Introduction The incomplete gamma function is given by Also, define Γ(α; ) = 1 with = G(α; ) = Z 0 Z 0 Z t α 1 e t dt, α > 0, >0 t α 1 e t dt, α >
Uniform Convergence of Fourier Series Michael Taylor
Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula
Mean-Variance Analysis
Mean-Variance Analysis Jan Schneider McCombs School of Business University of Texas at Austin Jan Schneider Mean-Variance Analysis Beta Representation of the Risk Premium risk premium E t [Rt t+τ ] R1
The Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
EE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
Example Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
Written Examination. Antennas and Propagation (AA ) April 26, 2017.
Written Examination Antennas and Propagation (AA. 6-7) April 6, 7. Problem ( points) Let us consider a wire antenna as in Fig. characterized by a z-oriented linear filamentary current I(z) = I cos(kz)ẑ
Reminders: linear functions
Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U
Commutative Monoids in Intuitionistic Fuzzy Sets
Commutative Monoids in Intuitionistic Fuzzy Sets S K Mala #1, Dr. MM Shanmugapriya *2 1 PhD Scholar in Mathematics, Karpagam University, Coimbatore, Tamilnadu- 641021 Assistant Professor of Mathematics,
Jesse Maassen and Mark Lundstrom Purdue University November 25, 2013
Notes on Average Scattering imes and Hall Factors Jesse Maassen and Mar Lundstrom Purdue University November 5, 13 I. Introduction 1 II. Solution of the BE 1 III. Exercises: Woring out average scattering
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ. Ψηφιακή Οικονομία. Διάλεξη 8η: Producer Behavior Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Ψηφιακή Οικονομία Διάλεξη 8η: Producer Behavior Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών Firm Behavior GOAL: Firms choose the maximum possible output (technological
ECE Spring Prof. David R. Jackson ECE Dept. Notes 2
ECE 634 Spring 6 Prof. David R. Jackson ECE Dept. Notes Fields in a Source-Free Region Example: Radiation from an aperture y PEC E t x Aperture Assume the following choice of vector potentials: A F = =
Introduction to the ML Estimation of ARMA processes
Introduction to the ML Estimation of ARMA processes Eduardo Rossi University of Pavia October 2013 Rossi ARMA Estimation Financial Econometrics - 2013 1 / 1 We consider the AR(p) model: Y t = c + φ 1 Y
Module 5. February 14, h 0min
Module 5 Stationary Time Series Models Part 2 AR and ARMA Models and Their Properties Class notes for Statistics 451: Applied Time Series Iowa State University Copyright 2015 W. Q. Meeker. February 14,
Econ 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1
Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test
An Introduction to Signal Detection and Estimation - Second Edition Chapter II: Selected Solutions
An Introduction to Signal Detection Estimation - Second Edition Chapter II: Selected Solutions H V Poor Princeton University March 16, 5 Exercise : The likelihood ratio is given by L(y) (y +1), y 1 a With
P AND P. P : actual probability. P : risk neutral probability. Realtionship: mutual absolute continuity P P. For example:
(B t, S (t) t P AND P,..., S (p) t ): securities P : actual probability P : risk neutral probability Realtionship: mutual absolute continuity P P For example: P : ds t = µ t S t dt + σ t S t dw t P : ds
Math 446 Homework 3 Solutions. (1). (i): Reverse triangle inequality for metrics: Let (X, d) be a metric space and let x, y, z X.
Math 446 Homework 3 Solutions. (1). (i): Reverse triangle inequalit for metrics: Let (X, d) be a metric space and let x,, z X. Prove that d(x, z) d(, z) d(x, ). (ii): Reverse triangle inequalit for norms:
Chapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
ΕΙΣΑΓΩΓΗ ΣΤΗ ΣΤΑΤΙΣΤΙΚΗ ΑΝΑΛΥΣΗ
ΕΙΣΑΓΩΓΗ ΣΤΗ ΣΤΑΤΙΣΤΙΚΗ ΑΝΑΛΥΣΗ ΕΛΕΝΑ ΦΛΟΚΑ Επίκουρος Καθηγήτρια Τµήµα Φυσικής, Τοµέας Φυσικής Περιβάλλοντος- Μετεωρολογίας ΓΕΝΙΚΟΙ ΟΡΙΣΜΟΙ Πληθυσµός Σύνολο ατόµων ή αντικειµένων στα οποία αναφέρονται
Nondifferentiable Convex Functions
Nondifferentiable Convex Functions DS-GA 1013 / MATH-GA 2824 Optimization-based Data Analysis http://www.cims.nyu.edu/~cfgranda/pages/obda_fall17/index.html Carlos Fernandez-Granda Applications Subgradients
6.3 Forecasting ARMA processes
122 CHAPTER 6. ARMA MODELS 6.3 Forecasting ARMA processes The purpose of forecasting is to predict future values of a TS based on the data collected to the present. In this section we will discuss a linear
Homework 3 Solutions
Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For
Chapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question 1 a) The cumulative distribution function of T conditional on N n is Pr (T t N n) Pr (max (X 1,..., X N ) t N n) Pr (max
Problem Set 3: Solutions
CMPSCI 69GG Applied Information Theory Fall 006 Problem Set 3: Solutions. [Cover and Thomas 7.] a Define the following notation, C I p xx; Y max X; Y C I p xx; Ỹ max I X; Ỹ We would like to show that C
SCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions
SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)
Ó³ Ÿ , º 2(131).. 105Ä ƒ. ± Ï,.. ÊÉ ±μ,.. Šμ ² ±μ,.. Œ Ì ²μ. Ñ Ò É ÉÊÉ Ö ÒÌ ² μ, Ê
Ó³ Ÿ. 2006.. 3, º 2(131).. 105Ä110 Š 537.311.5; 538.945 Œ ƒ ˆ ƒ Ÿ ˆŠ ˆ ƒ Ÿ ƒ ˆ œ ƒ Œ ƒ ˆ ˆ Š ˆ 4 ². ƒ. ± Ï,.. ÊÉ ±μ,.. Šμ ² ±μ,.. Œ Ì ²μ Ñ Ò É ÉÊÉ Ö ÒÌ ² μ, Ê ³ É É Ö μ ² ³ μ É ³ Í ² Ö Ê³ μ μ ³ É μ μ μ²ö
INTERVAL ARITHMETIC APPLIED TO STRUCTURAL DESIGN OF UNCERTAIN MECHANICAL SYSTEMS
INTERVAL ARITHMETIC APPLIED TO STRUCTURAL DESIGN OF UNCERTAIN MECHANICAL SYSTEMS O. Dessombz, F.Thouverez J-P. Laîné, L.Jézéquel Laboratoire de Tribologie et Dynamique des Systèmes UMR CNRS 5513 École
Sequent Calculi for the Modal µ-calculus over S5. Luca Alberucci, University of Berne. Logic Colloquium Berne, July 4th 2008
Sequent Calculi for the Modal µ-calculus over S5 Luca Alberucci, University of Berne Logic Colloquium Berne, July 4th 2008 Introduction Koz: Axiomatisation for the modal µ-calculus over K Axioms: All classical
Feasible Regions Defined by Stability Constraints Based on the Argument Principle
Feasible Regions Defined by Stability Constraints Based on the Argument Principle Ken KOUNO Masahide ABE Masayuki KAWAMATA Department of Electronic Engineering, Graduate School of Engineering, Tohoku University
➂ 6 P 3 ➀ 94 q ❸ ❸ q ❼ q ❿ P ❿ ➅ ➅ 3 ➁ ➅ 3 ➅ ❾ ❶ P 4 ➀ q ❺ q ❸ ❸ ➄ ❾➃ ❼ 2 ❿ ❹ 5➒ 3 ➀ 96 q ➀ 3 2 ❾ 2 ❼ ❸ ➄3 q ❸ ➆ q s 3 ➀ 94 q ➂ P ❺ 10 5 ➊ ➋➃ ❸ ❾ 3➃ ❼
P P P q r s t 1 2 34 5 P P 36 2 P 7 8 94 q r Pq 10 ❶ ❶ ❷10 ❹❸ ❸ 9 ❺ ❼❻ q ❽ ❾ 2 ❿ 2 ❼❻ ➀ ➁ ➂ ❿ 3➃ ➄ 94 ➁ ➅ ❽ ➆ ➇ ➉➈ ➊ ➋ ➌ ➊ ➍ ➎ ➋ ➏➃ ➃ q ❺➐ 8 ➄ q ❷ P ➑ P ➅ ➇ ❽ ➈➃ ➒➇ ➓ ➏ ➎ ➄ P q 96 5P q 4 ❿ ➅ ➇➃❽ ➈➃ ➇ ➓
( ) 2 and compare to M.
Problems and Solutions for Section 4.2 4.9 through 4.33) 4.9 Calculate the square root of the matrix 3!0 M!0 8 Hint: Let M / 2 a!b ; calculate M / 2!b c ) 2 and compare to M. Solution: Given: 3!0 M!0 8
Š Ÿ Š Ÿ Ÿ ˆ Œ ˆŠ -280
Ó³ Ÿ.. 2012.. 9, º 8.. 89Ä97 Š Ÿ Š Ÿ Ÿ ˆ Œ ˆŠ -280 ƒ. ƒ. ƒê²ó ±Ö,.. Ê, ƒ.. Š ³ÒÏ,.. Š ³ÒÏ,. ±μ Ñ Ò É ÉÊÉ Ö ÒÌ ² μ, Ê ³ É É Ö Ò μ±μî ÉμÉ Ö Ê ±μ ÖÕÐ Ö É ³ ÉÒ ³μ μ μ Éμ Ö - ÒÌ ±Í ³. ƒ.. ² μ Ñ μ μ É ÉÊÉ Ö
Parts Manual. Trio Mobile Surgery Platform. Model 1033
Trio Mobile Surgery Platform Model 1033 Parts Manual For parts or technical assistance: Pour pièces de service ou assistance technique : Für Teile oder technische Unterstützung Anruf: Voor delen of technische
Section 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Exercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.
Exercises 0 More exercises are available in Elementary Differential Equations. If you have a problem to solve any of them, feel free to come to office hour. Problem Find a fundamental matrix of the given
EE 570: Location and Navigation
EE 570: Location and Navigation INS Initialization Aly El-Osery Kevin Wedeward Electrical Engineering Department, New Mexico Tech Socorro, New Mexico, USA In Collaboration with Stephen Bruder Electrical
Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)
Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts
( y) Partial Differential Equations
Partial Dierential Equations Linear P.D.Es. contains no owers roducts o the deendent variables / an o its derivatives can occasionall be solved. Consider eamle ( ) a (sometimes written as a ) we can integrate
Areas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
APPENDICES APPENDIX A. STATISTICAL TABLES AND CHARTS 651 APPENDIX B. BIBLIOGRAPHY 677 APPENDIX C. ANSWERS TO SELECTED EXERCISES 679
APPENDICES APPENDIX A. STATISTICAL TABLES AND CHARTS 1 Table I Summary of Common Probability Distributions 2 Table II Cumulative Standard Normal Distribution Table III Percentage Points, 2 of the Chi-Squared
Areas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
w o = R 1 p. (1) R = p =. = 1
Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών ΗΥ-570: Στατιστική Επεξεργασία Σήµατος 205 ιδάσκων : Α. Μουχτάρης Τριτη Σειρά Ασκήσεων Λύσεις Ασκηση 3. 5.2 (a) From the Wiener-Hopf equation we have:
Monetary Policy Design in the Basic New Keynesian Model
Monetary Policy Design in the Basic New Keynesian Model Jordi Galí CREI, UPF and Barcelona GSE June 216 Jordi Galí (CREI, UPF and Barcelona GSE) Monetary Policy Design June 216 1 / 12 The Basic New Keynesian
Lecture 21: Scattering and FGR
ECE-656: Fall 009 Lecture : Scattering and FGR Professor Mark Lundstrom Electrical and Computer Engineering Purdue University, West Lafayette, IN USA Review: characteristic times τ ( p), (, ) == S p p
l 0 l 2 l 1 l 1 l 1 l 2 l 2 l 1 l p λ λ µ R N l 2 R N l 2 2 = N x i l p p R N l p N p = ( x i p ) 1 p i=1 l 2 l p p = 2 l p l 1 R N l 1 i=1 x 2 i 1 = N x i i=1 l p p p R N l 0 0 = {i x i 0} R
ACTA MATHEMATICAE APPLICATAE SINICA Nov., ( µ ) ( (
35 Þ 6 Ð Å Vol. 35 No. 6 2012 11 ACTA MATHEMATICAE APPLICATAE SINICA Nov., 2012 È ÄÎ Ç ÓÑ ( µ 266590) (E-mail: jgzhu980@yahoo.com.cn) Ð ( Æ (Í ), µ 266555) (E-mail: bbhao981@yahoo.com.cn) Þ» ½ α- Ð Æ Ä
The Probabilistic Method - Probabilistic Techniques. Lecture 7: The Janson Inequality
The Probabilistic Method - Probabilistic Techniques Lecture 7: The Janson Inequality Sotiris Nikoletseas Associate Professor Computer Engineering and Informatics Department 2014-2015 Sotiris Nikoletseas,
M p f(p, q) = (p + q) O(1)
l k M = E, I S = {S,..., S t } E S i = p i {,..., t} S S q S Y E q X S X Y = X Y I X S X Y = X Y I S q S q q p+q p q S q p i O q S pq p i O S 2 p q q p+q p q p+q p fp, q AM S O fp, q p + q p p+q p AM
Concrete Mathematics Exercises from 30 September 2016
Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)
b. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds!
MTH U341 urface Integrals, tokes theorem, the divergence theorem To be turned in Wed., Dec. 1. 1. Let be the sphere of radius a, x 2 + y 2 + z 2 a 2. a. Use spherical coordinates (with ρ a) to parametrize.
Takeaki Yamazaki (Toyo Univ.) 山崎丈明 ( 東洋大学 ) Oct. 24, RIMS
Takeaki Yamazaki (Toyo Univ.) 山崎丈明 ( 東洋大学 ) Oct. 24, 2017 @ RIMS Contents Introduction Generalized Karcher equation Ando-Hiai inequalities Problem Introduction PP: The set of all positive definite operators
Inverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in
Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in : tail in X, head in A nowhere-zero Γ-flow is a Γ-circulation such that
6.1. Dirac Equation. Hamiltonian. Dirac Eq.
6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2
TMA4115 Matematikk 3
TMA4115 Matematikk 3 Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet Trondheim Spring 2010 Lecture 12: Mathematics Marvellous Matrices Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet
Theorem 8 Let φ be the most powerful size α test of H
Testing composite hypotheses Θ = Θ 0 Θ c 0 H 0 : θ Θ 0 H 1 : θ Θ c 0 Definition 16 A test φ is a uniformly most powerful (UMP) level α test for H 0 vs. H 1 if φ has level α and for any other level α test
2?nom. Bacc. 2 nom. acc. S <u. >nom. 7acc. acc >nom < <
K+P K+P PK+ K+P - _+ l Š N K - - a\ Q4 Q + hz - I 4 - _+.P k - G H... /.4 h i j j - 4 _Q &\\ \\ ` J K aa\ `- c -+ _Q K J K -. P.. F H H - H - _+ 4 K4 \\ F &&. P H.4 Q+ 4 G H J + I K/4 &&& && F : ( -+..
Chapter 5, 6 Multiple Random Variables ENCS Probability and Stochastic Processes
Chapter 5, 6 Multiple Random Variables ENCS6161 - Probability and Stochastic Processes Concordia University ENCS6161 p.1/47 Vector Random Variables A vector r.v. X is a function X : S R n, where S is the
!"!# ""$ %%"" %$" &" %" "!'! " #$!
" "" %%"" %" &" %" " " " % ((((( ((( ((((( " %%%% & ) * ((( "* ( + ) (((( (, (() (((((* ( - )((((( )((((((& + )(((((((((( +. ) ) /(((( +( ),(, ((((((( +, 0 )/ (((((+ ++, ((((() & "( %%%%%%%%%%%%%%%%%%%(
2. Let H 1 and H 2 be Hilbert spaces and let T : H 1 H 2 be a bounded linear operator. Prove that [T (H 1 )] = N (T ). (6p)
![2. Let H 1 and H 2 be Hilbert spaces and let T : H 1 H 2 be a bounded linear operator. Prove that [T (H 1 )] = N (T ). (6p)](/thumbs/40/21364883.jpg "2. Let H 1 and H 2 be Hilbert spaces and let T : H 1 H 2 be a bounded linear operator. Prove that [T (H 1 )] = N (T ). (6p)")
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 2005-03-08 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
Mean bond enthalpy Standard enthalpy of formation Bond N H N N N N H O O O
Q1. (a) Explain the meaning of the terms mean bond enthalpy and standard enthalpy of formation. Mean bond enthalpy... Standard enthalpy of formation... (5) (b) Some mean bond enthalpies are given below.
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ. Ψηφιακή Οικονομία. Διάλεξη 7η: Consumer Behavior Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Ψηφιακή Οικονομία Διάλεξη 7η: Consumer Behavior Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών Τέλος Ενότητας Χρηματοδότηση Το παρόν εκπαιδευτικό υλικό έχει αναπτυχθεί
Quadratic Expressions
Quadratic Expressions. The standard form of a quadratic equation is ax + bx + c = 0 where a, b, c R and a 0. The roots of ax + bx + c = 0 are b ± b a 4ac. 3. For the equation ax +bx+c = 0, sum of the roots
CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Discontinuous Hermite Collocation and Diagonally Implicit RK3 for a Brain Tumour Invasion Model
1 Discontinuous Hermite Collocation and Diagonally Implicit RK3 for a Brain Tumour Invasion Model John E. Athanasakis Applied Mathematics & Computers Laboratory Technical University of Crete Chania 73100,
Dong Liu State Key Laboratory of Particle Detection and Electronics University of Science and Technology of China
Dong Liu State Key Laboratory of Particle Detection and Electronics University of Science and Technology of China ISSP, Erice, 7 Outline Introduction of BESIII experiment Motivation of the study Data sample
P ƒ.. Š ³ÒÏ,.. Š ³ÒÏ,.. ± ˆ ŒˆŠˆ Š ˆŠ
P9-2008-53 ƒ.. Š ³ÒÏ,.. Š ³ÒÏ,.. ± ˆ ŒˆŠˆ Š ˆŠ ˆ Œ MATLAB Š ³ÒÏ ƒ.., Š ³ÒÏ.., ±.. P9-2008-53 Î ÉÒ ³ ± Êα Í ±²μÉ μ Ì É ³ MATLAB É ÉÓ μ± μ ³μ μ ÉÓ ³ Ö Œ LAB ²Ö ÊÎ ÒÌ Î - Éμ Ë ± Ê ±μ É ², Î É μ É ²Ö μ Ö