Instanton Effects in ABJ(M) Theory
|
|
|
- Βηθζαθά Νέμεσις Αποστολίδης
- 5 χρόνια πριν
- Προβολές:
Transcript
1 Instanton Effects in ABJ(M) Theory Sanefumi Moriyama (NagoyaU KMI) [Fuji+Hirano+M 1106] [Hatsuda+M+Okuyama 1207, 1211, 1301] [HMO+Marino 1306] [HMO+Honda 1306] [Matsumoto+M 1310]
2 0. Introduction
3 Physics "Special" Background Highly Symmetric, Simple OR EVEN Solvable "General" Backgrounds By Perturbation Theory or Other Methods (Example) e βh with H=p 2 /2+q 2 /2+g q 4 /4
4 Perturbative String Theory Perturbative String Theories on 10D Flat Spacetime Perturbation Theory Amplitudes (Small Fluctuation) β Functions (Background EOM)...
5 Non Perturbatively Situation Changes Drastically! In Strong Coupling Limit, M Theory!! Most Symmetric Cases M Theory on 11D Flat Spacetime M Theory on AdS 7 4 x S M Theory on AdS 7 x S 4 Still Very Mysterious = Unsatisfied
6 M2 branes [Aharony Bergman Jafferis Maldacena] N=6 N6 Super Chern Simons Theory = M2 on C 4 /Z k [Witten, Pestun,...] Supersymmetric Observables : Matrix Integral Results: ABJM Matrix Model with Interesting Structure Extract As Much Physics As Possible
7 Contents 1. Quick Review of M Theory What did we learn two decades ago? 2. ABJ(M) Wilson Loop Fractional Branes 3. Exact Instanton Expansion String Democracy
8 1. Quick Review of M Theory
9 M is for Mother 5 Consistent tstring Theories in 10D Het E8xE8 IIA Het SO(32) IIB I
10 M is for Mother 5 Consistent tstring Theories in 10D 5 Vacua of A Unique String Theory Het E8xE8 IIA Het SO(32) String Duality IIB D brane I
11 M is for Mother Het E8xE8 M (11D) Strong Coupling Limit IIA 10D Het SO(32) IIB I
12 M is for Membrane Fundamental Solitonic i M2 brane M5 brane Lessons String D2 brane String Theory NOT Just "a theory of strings" Only Consistent withbranes But, Branes also as "Composites of Strings" (from SFT viewpoints)
13 M is for Mystery DOF N 2 for N D branes Described by Matrix
14 M is for Mystery M2 brane M2 brane DOF N 3/2 /N 3 for N M2 /M5 branes
15 To Summarize we only knew little on To Summarize, we only knew little on "What M Theory Is" so far!
16 2. ABJ(M) Wilson Loop
17 ABJ(M) Theory [Aharony Bergman Jefferis (Maldacena)] N=6 N6Chern Simons matter matter Theory adjoint U(N 1 ) U(N 2 ) bifundamental adjoint Min(N 1,N 2 )x M2 & N 2 N 1 x fractional M2 on R 8 / Z k
18 Fractional brane & Wilson loop s Y k (N 1,N 2 ) Partition Function OR VEV of Half BPS Wilson Loop in Rep Y on Min(N 1,NN 2 ) x M2 & N 2 NN 1 x Fractional lm2 s Y k (N 1,N 2 ) = DA exp( S ABJ(M) [A, ]) Tr P exp A μ dx μ + = After Localization Techniques,...
19 ABJ(M) Matrix Model Physically,
20 ABJ(M) Matrix Model [Kapustin Willet Yaakov, Drukker Trancanelli, Marino Putrov] (Fresnel Integral) Character of U(N 1 N 2 )in Rep Y (Classified by Young Diagram Y) = Supersymmetric Schur Poly.
21 ABJ(M) Matrix Model [Kapustin Willet Yaakov, Drukker Trancanelli, Marino Putrov] (Fresnel Integral) Character of U(N 1 N 2 )in Rep Y (Classified by Young Diagram Y) = Supersymmetric Schur Poly.
22 Invariant Measure
23 Invariant Measure
24 Invariant Measure Chern Simons (q deform) U(N 1 ) Inv
25 Invariant Measure Gaussian Matrix Model H: N x N Hermitian Matrix H: N 1 x N 1 Hermitian Matrix Gauge Sym = U(N 1 )
26 Invariant Measure Chern Simons (q deform) U(N 1 ) Inv
27 Invariant Measure Chern Simons (q deform) U(N 1 N 2 ) Inv Supersymmetry U(N 1 ) Inv
28 Invariant Measure Chern Simons (q deform) ABJ(M) Matrix Model GaussianMatrix Model + Supersymmetry Deformation + Chern Simons Deformation U(N 1 N 2 ) Inv Supersymmetry U(N 1 ) Inv
29 Message ABJ(M) Matrix Model, as Fundamental as Gaussian Model or Chern Simons Model.
30 Grand Canonical Ensemble Without Loss of Generality, M=N 2 N 1 0, k>0 [s ( ) Y ] GC k,m(z) = N=0 s Y k (N,N+M) z N Regarding ABJ(M) Partition Function as PF of N Particle Fermi Gas System [Marino Putrov] s GC M(z) = GC k GC Y k,m [s Y ] k,m(z) / [1] k,0(z)
31 where and H p,q (z) = Theorem [Hatsuda Honda M Okuyama, Honda Matsumoto M] M] s GC Y k,m M(z) ( ) = det (M+r)x(M+r) H p,q( (z) E (1 + z 1 l p QP) 1 E M+q 1 z E (1 + z QP) 1 Q E lp aq M (Q) =... (P) =... (E j ) =... l p :... a q :... (M = N 2 N 1 ) (1 q M) (1 q M r)
![where and H p,q (z) = Theorem [Hatsuda Honda M Okuyama, Honda Matsumoto M] M] s GC Y k,m M(z) ( ) = det (M+r)x(M+r) H p,q( (z) E (1 + z 1 l p QP) 1 E M+q 1 z E (1 + z QP) 1 Q E lp aq M (Q) ν,μ =](/docs-images/89/98758805/images/32-1.jpg "[2cosh(ν μ)/2] 1 (P) μ,ν = [2cosh(μ ν)/2] 1 (E = e (j+1/2)ν j ) ν (Q) ν,μ, (P) μ,ν as Matrix & (E j ) ν as Vector (M = N 2 N 1 ) (1 q M) (1 q M r) with Continuous o Indices μ, ν Matrix Multiplication")
32 where and H p,q (z) = Theorem [Hatsuda Honda M Okuyama, Honda Matsumoto M] M] s GC Y k,m M(z) ( ) = det (M+r)x(M+r) H p,q( (z) E (1 + z 1 l p QP) 1 E M+q 1 z E (1 + z QP) 1 Q E lp aq M (Q) ν,μ = [2cosh(ν μ)/2] 1 (P) μ,ν = [2cosh(μ ν)/2] 1 (E = e (j+1/2)ν j ) ν (Q) ν,μ, (P) μ,ν as Matrix & (E j ) ν as Vector (M = N 2 N 1 ) (1 q M) (1 q M r) with Continuous o Indices μ, ν Matrix Multiplication = Integration with Dμ, Dν
33 where H p,q (z) = Theorem [Hatsuda Honda M Okuyama, Honda Matsumoto M] M] s GC Y k,m M(z) ( ) = det (M+r)x(M+r) H p,q( (z) E (1 + z 1 l p QP) 1 E M+q 1 z E (1 + z QP) 1 Q E lp aq M (M = N 2 N 1 ) (1 q M) (1 q M r) l p : p th leg length a q : q th arm length
34 Frobenius Symbol (a 1 a 2 a r l 1 l 2 l r+m ) U(N) or U(N N) U(N N+3) ( ) (3,2,0 9,7,5,4,2,1) or ( ) (6,5,3,2 6,4,2,1) ( 1, 2, 3,3,2,0 9,7,5,4,2,1) )
35 where H p,q (z) = Theorem [Hatsuda Honda M Okuyama, Honda Matsumoto M] M] s GC Y k,m M(z) ( ) = det (M+r)x(M+r) H p,q( (z) E (1 + z 1 l p QP) 1 E M+q 1 z E (1 + z QP) 1 Q E lp aq M (M = N 2 N 1 ) (1 q M) (1 q M r) leg # arm if not distinguishing between (1+zQP) 1 or z(1+zqp) 1 Q
36 Example GC k,m=3 det 1 # 9 1 # 7 1 # 5 1 # 4 1 # 2 1 # 1 2 # 9 2 # 7 2 # 5 2 # 4 2 # 2 2 # 1 3 # 9 3 # 7 3 # 5 3 # 4 3 # 2 3 # 1 3 # 9 3 # 7 3 # 5 3 # 4 3 # 2 3 # 1 2 # 9 2 # 7 2 # 5 2 # 4 2 # 2 2 # 1 0 # 9 0 # 7 0 # 5 0 # 4 0 # 2 0 # 1
37 Especially, ABJM Wilson loop det " General Representation = det Hook Representations "
38 Especially, ABJM Wilson loop Without VEVs,... det "(General Representation) = det (Hook Representations)" Giambelli Formula Giambelli Formula A Well Known Classical Formula in Mathematics
39 Especially, ABJM Wilson loop What we have proved... Possible to Put... GC k,m In Giambelli Formula "Giambelli Compatibility" for ABJM & "(Generalized) Giambelli Compatibility" for ABJ
40 Representation as Fermion Representation Y as Fermion Excitation (Example) Hook Representation Hook Representation Fundamental Excitation
41 Representation as Fermion Hook Representation = Fundamental Excitation General Representation = Solitonic Excitaion " General Representation = det Hook Representations " " Solitonic Excitation = det Fundamental Excitation "
42 Fermion as String String Fluctuation Wilson loop in Fundamental Representation Fermion as String
43 Especially, Fractional brane ABJ Without Wilson Loop Insertion GC 1 # 2 1 # 1 1 # 0 det 2 # 2 2 # 1 2 # 0 k,m=3 3 # 2 3 # 1 3 # 0 ABJ Partition Function In terms of "Hook" Fractional Branes from Fundamental Strings?
44 Break Summary So far Giambelli Compatibility for ABJM Wilson loop Generalized Giambelli Compatibility for ABJ Fractional Brane as String Composite Hereafter Exact Instanton Expansion
45 3. Exact Instanton Expansion
46 ABJM Theory [Aharony Bergman Jefferis Maldacena] N=6 N6Chern Simons matter matter Theory adjoint U(N) bifundamental U(N) adjoint N x M2 on R 8 / Z k (= R + x S 7 / Z k )
47 Pictorially S 7 / Z k S 7 / Z k k CP 3 x S 1
48 Shorthand Notation ABJM Partition Function ABJM Grand Potential Z(N) = 1 k (N,N) N) exp J(log z)= [1] GC k,0(z)
49 Developments Free Energy F(N)= Log Z(N)in large N Limit Perturbative ti Sum F(N) N 3/2 Z(N) ( ) = Ai[N] ( exp N 3/2 ) [Drukker Marino Putrov] [Fuji Hirano M, KEK, Marino Putrov]
50 Developments (Cont'd) Worldsheet Instanton (F1 wrapping CP 1 CP 3 ) [Drukker Marino Putrov, Hatsuda M Okuyama] Membrane Instanton (D2 wrapping RP 3 CP 3 ) Bound dstatet [Drukker Marino Putrov, Marino Putrov, Hatsuda M Okuyama, Calvo Marino] (Basically From Numerical Studies) [Hatsuda M Okuyama]
51 All Explicitly In Topological Strings [Hatsuda M Okuyama, Hatsuda M Marino Okuyama] Marino J(μ)=J pert (μ eff )+J WS (μ eff )+J MB (μ eff ) J pert (μ)=cμ 3 /3+Bμ+A J WS (μ eff )=F T top (T eff 1,T eff 2,λ) J MB (μ eff )=(2πi) 1 λ [λf NS (T 1 eff /λ,t 2 eff /λ,1/λ)] F top (T 1,T 2,τ) =... T eff 1 =4μ eff /k iπ F NS (T 1,TT 2,τ) =... T eff 2 =4μ eff /k+iπ C=2/π 2 k, B=..., A=... λ=2/k μ ( 1) k/2 2e 2μ 4F (1,1,3/2,3/2;2,2,2;( 1) k/2 16e 2μ ) k=even μ eff 4 3 = μ+e 4μ 4F 3 (1,1,3/2,3/2;2,2,2; 16e 4μ ) k=odd
52 All Explicitly In Topological Strings [Hatsuda M Okuyama, Hatsuda M Marino Okuyama] Marino J(μ)=J pert (μ eff )+J WS (μ eff )+J MB (μ eff ) J pert (μ)=cμ 3 /3+Bμ+A J WS (μ eff )=F T top (T eff 1,T eff 2,λ) J MB (μ eff )=(2πi) 1 λ [λf NS (T 1 eff /λ,t 2 eff /λ,1/λ)] F(T 1,T 2,τ 1,τ 2 ): Free Energy of Refined Top Strings T 1,T 2 : Kahler Moduli τ 1,τ 2 : Coupling Constants Topological Limit F top (T 1,T 2,τ) = lim τ1 τ,τ 2 τ F(T 1,T 2,τ 1,τ 2 ) NS Limit F NS (T 1,T 2,τ) = lim τ1 τ,τ 2 0 2πiτ 2 F(T 1,T 2,τ 1,τ 2 )
53 All Explicitly In Topological Strings [Hatsuda M Okuyama, Hatsuda M Marino Okuyama] Marino J(μ)=J pert (μ eff )+J WS (μ eff )+J MB (μ eff ) J pert (μ)=cμ 3 /3+Bμ+A J WS (μ eff )=F T top (T eff 1,T eff 2,λ) J MB (μ eff )=(2πi) 1 λ [λf NS (T 1 eff /λ,t 2 eff /λ,1/λ)] F(T 1,T 2,τ 1,τ 2 ) = jl,j R n d1,d 2 N jl,j R d 1,d 2 χ n(d T +d T ) jl (q L ) χ jr (q R ) e /[n(q 1 n/2 q 1 n/2 )(q 2 n/2 q 2 n/2 )] N jl j R d 1,d 2 :BPS Index of local P 1 x P 1 q 1 =e 2πiτ 1 q 2 =e 2πiτ 2 q L =e πi(τ 1 τ 2 ) q R =e πi(τ 1 +τ 2 ) j L,j R (Gopakumar Vafa or Gromov Witten invariants)
54 Methods 't Hooft Expansion [Drukker Marino Putrov] Perturbative WKB Expansion [Marino Putrov] Worldsheet Instanton Membrane Instanton Numerical Studies Cancellation Mechanism [Hatsuda M Okuyama]
55 Numerical Studies for k = 1, 2, 3, 4, 6 Compute Exact Values of Z(N) up to N max Read off Exact Values of J(μ) from Z(N) Numerically, After Subtracting Perturbative and Major Instanton Contribution Compare with Worldsheet Instanton
56 Instanton Effects Non Perturbative Part of Grand Potential J(μ) J k=1 (μ) = [#μ 2 +#μ+#]e 4μ + [#μ 2 +#μ+#]e 8μ + [#μ 2 +#μ+#]e 12μ +... J k=2 (μ) = [#μ 2 +#μ+#]e 2μ + [#μ 2 +#μ+#]e 4μ + [#μ 2 +#μ+#]e 6μ +... J k=3 (μ) = [#]e [] 4μ/3 + [#]e [] 8μ/3 + [#μ μ 2 +#μ+#]eμ 4μ +... J k=4 (μ) = [#]e μ + [#μ 2 +#μ+#]e 2μ + [#]e 3μ J k=6 (μ) = [#]e 2μ/3 + [#]e 4μ/3 + [#μ 2 +#μ+#]e 2μ +...
57 Instanton Effects Non Perturbative Part of Grand Potential J(μ) J k=1 (μ) = [#μ 2 +#μ+#]e 4μ + [#μ 2 +#μ+#]e 8μ + [#μ 2 +#μ+#]e 12μ +... J k=2 (μ) = [#μ 2 +#μ+#]e 2μ + [#μ 2 +#μ+#]e 4μ + [#μ 2 +#μ+#]e 6μ +... J k=3 (μ) = [#]e [] 4μ/3 + [#]e [] 8μ/3 + [#μ μ 2 +#μ+#]eμ 4μ +... J k=4 (μ) = [#]e μ + [#μ 2 +#μ+#]e 2μ + [#]e 3μ J k=6 (μ) = [#]e 2μ/3 + [#]e 4μ/3 + [#μ 2 +#μ+#]e 2μ +... WS(1) WS(2) WS(3)
58 Instanton Effects Worldsheet Instanton J k=1 (μ) = [#μ 2 +#μ+#]e 4μ + [#μ 2 +#μ+#]e 8μ + [#μ 2 +#μ+#]e 12μ +... J k=2 (μ) = [#μ 2 +#μ+#]e 2μ + [#μ 2 +#μ+#]e 4μ + [#μ 2 +#μ+#]e 6μ +... J k=3 (μ) = [#]e [] 4μ/3 + [#]e [] 8μ/3 + [#μ μ 2 +#μ+#]eμ 4μ +... J k=4 (μ) = [#]e μ + [#μ 2 +#μ+#]e 2μ + [#]e 3μ +... Match well with Topological String... J k=6 (μ) = [#]e 2μ/3 + [#]e 4μ/3 + [#μ 2 +#μ+#]e 2μ +... Prediction of WS WS(1) WS(2) WS(3)
59 Instanton Effects Worldsheet Instanton, Divergent at Certain k J k=1 (μ) = [#μ 2 +#μ+#]e 4μ + [#μ 2 +#μ+#]e 8μ + [#μ 2 +#μ+#]e 12μ +... J k=2 (μ) = [#μ 2 +#μ+#]e 2μ + [#μ 2 +#μ+#]e 4μ + [#μ 2 +#μ+#]e 6μ +... J k=3 (μ) = [#]e [] 4μ/3 + [#]e [] 8μ/3 + [#μ μ 2 +#μ+#]eμ 4μ +... J k=4 (μ) = [#]e μ + [#μ 2 +#μ+#]e 2μ + [#]e 3μ +... Match well with Topological String... J k=6 (μ) = [#]e 2μ/3 + [#]e 4μ/3 + [#μ 2 +#μ+#]e 2μ +... Prediction of WS WS(1) WS(2) WS(3)
60 Instanton Effects Worldsheet Instanton, Divergent at Certain k Divergence Cancelled by Membrane Instanton J k=1 (μ) = [#μ 2 +#μ+#]e 4μ + [#μ 2 +#μ+#]e 8μ + [#μ 2 +#μ+#]e 12μ +... J k=2 (μ) = [#μ 2 +#μ+#]e 2μ + [#μ 2 +#μ+#]e 4μ + [#μ 2 +#μ+#]e 6μ +... J k=3 (μ) = [#]e [] 4μ/3 + [#]e [] 8μ/3 + [#μ μ 2 +#μ+#]eμ 4μ +... k=3 MB(2) J k=4 (μ) = [#]e μ + [#μ 2 +#μ+#]e 2μ + [#]e 3μ +... Match well with Topological String... J k=6 (μ) = [#]e 2μ/3 + [#]e 4μ/3 + [#μ 2 +#μ+#]e 2μ +... Prediction of WS WS(1) WS(2) WS(3) MB(1)
61 Divergence Cancellation Mechanism Aesthetically Reproducing the Lessons String Theory, Not Just 'a theory of strings' Practically Helpful in Determining Membrane Instanton
62 An Incorrect but Suggestive Interpretation Worldsheet Inst S 7 / Z k 1 Instanton k InstantonI Off Fixed Pt cf: Twisted Sectors in String Orbifold
63 Cancellation New Branch in WS inst Divergence Cancelled by MB Inst
64 Compact Moduli Space? Compactified by Membrane Instanton NonPerturbatively!? Perturbative WorldSheet Instanton Moduli
65 Another Implication J(μ)=J pert (μ eff )+J WS (μ eff )+J MB (μ eff ) J pert (μ)=cμ 3 /3+Bμ+A J WS (μ eff )=F T top (T eff 1,T eff 2,λ) J MB (μ eff )=(2πi) 1 λ [λf NS (T 1 eff /λ,t 2 eff /λ,1/λ)] F(T 1,T 2,τ 1,τ 2 ) = jl,j R n d1,d 2 N jl,j R d 1,d 2 χ n(d T +d T ) jl (q L ) χ jr (q R ) e /[n(q 1 n/2 q 1 n/2 )(q 2 n/2 q 2 n/2 )] NonPerturbative Topological Strings on General Background by Requiring Divergence Cancellation [Hatsuda Marino M Okuyama]
66 Short Summary Summary Explicit Form of Membrane Instanton Exact Large N Expansion of ABJM Partition Function Divergence Cancellation Moduli Space of Membrane?
67 Summary & Further Directions ABJM Partition Function Exact Large N Expansion Divergence Cancellation Fractional lmembrane from Wilson Loop Generalization for M2 Orbifolds, Orientifolds, Ellipsoid/Squashed S 3 Implication of Cancellation for M5 Exploring Moduli Space of M theory
68 Thank you for your attention!
Higher spin gauge theories and their CFT duals
Higher spin gauge theories and their CFT duals E-mail: hikida@phys-h.keio.ac.jp 2 AdS Vasiliev AdS/CFT 4 Vasiliev 3 O(N) 3 Vasiliev 2 W N 1 AdS/CFT g µν Vasiliev AdS [1] AdS/CFT anti-de Sitter (AdS) (CFT)
DiracDelta. Notations. Primary definition. Specific values. General characteristics. Traditional name. Traditional notation
DiracDelta Notations Traditional name Dirac delta function Traditional notation x Mathematica StandardForm notation DiracDeltax Primary definition 4.03.02.000.0 x Π lim ε ; x ε0 x 2 2 ε Specific values
Fourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)
3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
Space-Time Symmetries
Chapter Space-Time Symmetries In classical fiel theory any continuous symmetry of the action generates a conserve current by Noether's proceure. If the Lagrangian is not invariant but only shifts by a
g-selberg integrals MV Conjecture An A 2 Selberg integral Summary Long Live the King Ole Warnaar Department of Mathematics Long Live the King
Ole Warnaar Department of Mathematics g-selberg integrals The Selberg integral corresponds to the following k-dimensional generalisation of the beta integral: D Here and k t α 1 i (1 t i ) β 1 1 i
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
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
Dr. D. Dinev, Department of Structural Mechanics, UACEG
Lecture 4 Material behavior: Constitutive equations Field of the game Print version Lecture on Theory of lasticity and Plasticity of Dr. D. Dinev, Department of Structural Mechanics, UACG 4.1 Contents
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all
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
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
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
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 :
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)
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
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
Three coupled amplitudes for the πη, K K and πη channels without data
Three coupled amplitudes for the πη, K K and πη channels without data Robert Kamiński IFJ PAN, Kraków and Łukasz Bibrzycki Pedagogical University, Kraków HaSpect meeting, Kraków, V/VI 216 Present status
DETERMINATION OF DYNAMIC CHARACTERISTICS OF A 2DOF SYSTEM. by Zoran VARGA, Ms.C.E.
DETERMINATION OF DYNAMIC CHARACTERISTICS OF A 2DOF SYSTEM by Zoran VARGA, Ms.C.E. Euro-Apex B.V. 1990-2012 All Rights Reserved. The 2 DOF System Symbols m 1 =3m [kg] m 2 =8m m=10 [kg] l=2 [m] E=210000
CRASH COURSE IN PRECALCULUS
CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter
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.
Revisiting the S-matrix approach to the open superstring low energy eective lagrangian
Revisiting the S-matrix approach to the open superstring low energy eective lagrangian IX Simposio Latino Americano de Altas Energías Memorial da América Latina, São Paulo. Diciembre de 2012. L. A. Barreiro
Laplace Expansion. Peter McCullagh. WHOA-PSI, St Louis August, Department of Statistics University of Chicago
Laplace Expansion Peter McCullagh Department of Statistics University of Chicago WHOA-PSI, St Louis August, 2017 Outline Laplace approximation in 1D Laplace expansion in 1D Laplace expansion in R p Formal
12. Radon-Nikodym Theorem
Tutorial 12: Radon-Nikodym Theorem 1 12. Radon-Nikodym Theorem In the following, (Ω, F) is an arbitrary measurable space. Definition 96 Let μ and ν be two (possibly complex) measures on (Ω, F). We say
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
C.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Matrices and vectors. Matrix and vector. a 11 a 12 a 1n a 21 a 22 a 2n A = b 1 b 2. b m. R m n, b = = ( a ij. a m1 a m2 a mn. def
Matrices and vectors Matrix and vector a 11 a 12 a 1n a 21 a 22 a 2n A = a m1 a m2 a mn def = ( a ij ) R m n, b = b 1 b 2 b m Rm Matrix and vectors in linear equations: example E 1 : x 1 + x 2 + 3x 4 =
9.09. # 1. Area inside the oval limaçon r = cos θ. To graph, start with θ = 0 so r = 6. Compute dr
9.9 #. Area inside the oval limaçon r = + cos. To graph, start with = so r =. Compute d = sin. Interesting points are where d vanishes, or at =,,, etc. For these values of we compute r:,,, and the values
The Spiral of Theodorus, Numerical Analysis, and Special Functions
Theo p. / The Spiral of Theodorus, Numerical Analysis, and Special Functions Walter Gautschi wxg@cs.purdue.edu Purdue University Theo p. 2/ Theodorus of ca. 46 399 B.C. Theo p. 3/ spiral of Theodorus 6
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
Notations. Primary definition. Specific values. General characteristics. Traditional name. Traditional notation. Mathematica StandardForm notation
KelvinKei Notations Traditional name Kelvin function of the second kind Traditional notation kei Mathematica StandardForm notation KelvinKei Primary definition 03.5.0.000.0 kei kei 0 Specific values Values
Gauge-Stringy Instantons
Gauge-Stringy Instantons Parsa Hossein Ghorbani Institute for Research in Fundamental Sciences (IPM) School of Particles and Accelerators INFN Sezione di Napoli July 2013 Parsa Hossein Ghorbani Gauge-Stringy
Lecture 2. Soundness and completeness of propositional logic
Lecture 2 Soundness and completeness of propositional logic February 9, 2004 1 Overview Review of natural deduction. Soundness and completeness. Semantics of propositional formulas. Soundness proof. Completeness
Απόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.
Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action
Math221: HW# 1 solutions
Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
Arithmetical applications of lagrangian interpolation. Tanguy Rivoal. Institut Fourier CNRS and Université de Grenoble 1
Arithmetical applications of lagrangian interpolation Tanguy Rivoal Institut Fourier CNRS and Université de Grenoble Conference Diophantine and Analytic Problems in Number Theory, The 00th anniversary
If we restrict the domain of y = sin x to [ π, π ], the restrict function. y = sin x, π 2 x π 2
![If we restrict the domain of y = sin x to [ π, π ], the restrict function. y = sin x, π 2 x π 2](/thumbs/53/31933086.jpg "If we restrict the domain of y = sin x to [ π, π ], the restrict function. y = sin x, π 2 x π 2")
Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the
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
ω ω ω ω ω ω+2 ω ω+2 + ω ω ω ω+2 + ω ω+1 ω ω+2 2 ω ω ω ω ω ω ω ω+1 ω ω2 ω ω2 + ω ω ω2 + ω ω ω ω2 + ω ω+1 ω ω2 + ω ω+1 + ω ω ω ω2 + ω
0 1 2 3 4 5 6 ω ω + 1 ω + 2 ω + 3 ω + 4 ω2 ω2 + 1 ω2 + 2 ω2 + 3 ω3 ω3 + 1 ω3 + 2 ω4 ω4 + 1 ω5 ω 2 ω 2 + 1 ω 2 + 2 ω 2 + ω ω 2 + ω + 1 ω 2 + ω2 ω 2 2 ω 2 2 + 1 ω 2 2 + ω ω 2 3 ω 3 ω 3 + 1 ω 3 + ω ω 3 +
ExpIntegralE. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation
ExpIntegralE Notations Traditional name Exponential integral E Traditional notation E Mathematica StandardForm notation ExpIntegralE, Primary definition 06.34.0.000.0 E t t t ; Re 0 Specific values Specialied
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,
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
If we restrict the domain of y = sin x to [ π 2, π 2
Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the
Code Breaker. TEACHER s NOTES
TEACHER s NOTES Time: 50 minutes Learning Outcomes: To relate the genetic code to the assembly of proteins To summarize factors that lead to different types of mutations To distinguish among positive,
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
Tridiagonal matrices. Gérard MEURANT. October, 2008
Tridiagonal matrices Gérard MEURANT October, 2008 1 Similarity 2 Cholesy factorizations 3 Eigenvalues 4 Inverse Similarity Let α 1 ω 1 β 1 α 2 ω 2 T =......... β 2 α 1 ω 1 β 1 α and β i ω i, i = 1,...,
Higher Derivative Gravity Theories
Higher Derivative Gravity Theories Black Holes in AdS space-times James Mashiyane Supervisor: Prof Kevin Goldstein University of the Witwatersrand Second Mandelstam, 20 January 2018 James Mashiyane WITS)
Non-Gaussianity from Lifshitz Scalar
COSMO/CosPA 200 September 27, 200 Non-Gaussianity from Lifshitz Scalar Takeshi Kobayashi (Tokyo U.) based on: arxiv:008.406 with Keisuke Izumi, Shinji Mukohyama Lifshitz scalars with z=3 obtain super-horizon,
Elements of Information Theory
Elements of Information Theory Model of Digital Communications System A Logarithmic Measure for Information Mutual Information Units of Information Self-Information News... Example Information Measure
Section 7.6 Double and Half Angle Formulas
09 Section 7. Double and Half Angle Fmulas To derive the double-angles fmulas, we will use the sum of two angles fmulas that we developed in the last section. We will let α θ and β θ: cos(θ) cos(θ + θ)
Συστήματα Διαχείρισης Βάσεων Δεδομένων
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Συστήματα Διαχείρισης Βάσεων Δεδομένων Φροντιστήριο 9: Transactions - part 1 Δημήτρης Πλεξουσάκης Τμήμα Επιστήμης Υπολογιστών Tutorial on Undo, Redo and Undo/Redo
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
( ) 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
Υπολογιστική Φυσική Στοιχειωδών Σωματιδίων
Υπολογιστική Φυσική Στοιχειωδών Σωματιδίων Όρια Πιστότητας (Confidence Limits) 2/4/2014 Υπολογ.Φυσική ΣΣ 1 Τα όρια πιστότητας -Confidence Limits (CL) Tα όρια πιστότητας μιας μέτρησης Μπορεί να αναφέρονται
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
Finite Field Problems: Solutions
Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The
Review Test 3. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Review Test MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. 1) sin - 11π 1 1) + - + - - ) sin 11π 1 ) ( -
Notations. Primary definition. Specific values. General characteristics. Series representations. Traditional name. Traditional notation
Pi Notations Traditional name Π Traditional notation Π Mathematica StandardForm notation Pi Primary definition.3... Π Specific values.3.3.. Π 3.5965358979338663383795889769399375589795937866868998683853
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
SPECIAL FUNCTIONS and POLYNOMIALS
SPECIAL FUNCTIONS and POLYNOMIALS Gerard t Hooft Stefan Nobbenhuis Institute for Theoretical Physics Utrecht University, Leuvenlaan 4 3584 CC Utrecht, the Netherlands and Spinoza Institute Postbox 8.195
6. MAXIMUM LIKELIHOOD ESTIMATION
6 MAXIMUM LIKELIHOOD ESIMAION [1] Maximum Likelihood Estimator (1) Cases in which θ (unknown parameter) is scalar Notational Clarification: From now on, we denote the true value of θ as θ o hen, view θ
Congruence Classes of Invertible Matrices of Order 3 over F 2
International Journal of Algebra, Vol. 8, 24, no. 5, 239-246 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ija.24.422 Congruence Classes of Invertible Matrices of Order 3 over F 2 Ligong An and
Heisenberg Uniqueness pairs
Heisenberg Uniqueness pairs Philippe Jaming Bordeaux Fourier Workshop 2013, Renyi Institute Joint work with K. Kellay Heisenberg Uniqueness Pairs µ : finite measure on R 2 µ(x, y) = R 2 e i(sx+ty) dµ(s,
Biostatistics for Health Sciences Review Sheet
Biostatistics for Health Sciences Review Sheet http://mathvault.ca June 1, 2017 Contents 1 Descriptive Statistics 2 1.1 Variables.............................................. 2 1.1.1 Qualitative........................................
Graded Refractive-Index
Graded Refractive-Index Common Devices Methodologies for Graded Refractive Index Methodologies: Ray Optics WKB Multilayer Modelling Solution requires: some knowledge of index profile n 2 x Ray Optics for
Every set of first-order formulas is equivalent to an independent set
Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent
General 2 2 PT -Symmetric Matrices and Jordan Blocks 1
General 2 2 PT -Symmetric Matrices and Jordan Blocks 1 Qing-hai Wang National University of Singapore Quantum Physics with Non-Hermitian Operators Max-Planck-Institut für Physik komplexer Systeme Dresden,
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,
Lecture 10 - Representation Theory III: Theory of Weights
Lecture 10 - Representation Theory III: Theory of Weights February 18, 2012 1 Terminology One assumes a base = {α i } i has been chosen. Then a weight Λ with non-negative integral Dynkin coefficients Λ
A Lambda Model Characterizing Computational Behaviours of Terms
A Lambda Model Characterizing Computational Behaviours of Terms joint paper with Silvia Ghilezan RPC 01, Sendai, October 26, 2001 1 Plan of the talk normalization properties inverse limit model Stone dualities
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
What happens when two or more waves overlap in a certain region of space at the same time?
Wave Superposition What happens when two or more waves overlap in a certain region of space at the same time? To find the resulting wave according to the principle of superposition we should sum the fields
CHAPTER 48 APPLICATIONS OF MATRICES AND DETERMINANTS
CHAPTER 48 APPLICATIONS OF MATRICES AND DETERMINANTS EXERCISE 01 Page 545 1. Use matrices to solve: 3x + 4y x + 5y + 7 3x + 4y x + 5y 7 Hence, 3 4 x 0 5 y 7 The inverse of 3 4 5 is: 1 5 4 1 5 4 15 8 3
Wishart α-determinant, α-hafnian
Wishart α-determinant, α-hafnian (, JST CREST) (, JST CREST), Wishart,. ( )Wishart,. determinant Hafnian analogue., ( )Wishart,. 1 Introduction, Wishart. p ν M = (µ 1,..., µ ν ) = (µ ij ) i=1,...,p p p
Divergence for log concave functions
Divergence or log concave unctions Umut Caglar The Euler International Mathematical Institute June 22nd, 2013 Joint work with C. Schütt and E. Werner Outline 1 Introduction 2 Main Theorem 3 -divergence
A Bonus-Malus System as a Markov Set-Chain. Małgorzata Niemiec Warsaw School of Economics Institute of Econometrics
A Bonus-Malus System as a Markov Set-Chain Małgorzata Niemiec Warsaw School of Economics Institute of Econometrics Contents 1. Markov set-chain 2. Model of bonus-malus system 3. Example 4. Conclusions
10/3/ revolution = 360 = 2 π radians = = x. 2π = x = 360 = : Measures of Angles and Rotations
//.: Measures of Angles and Rotations I. Vocabulary A A. Angle the union of two rays with a common endpoint B. BA and BC C. B is the vertex. B C D. You can think of BA as the rotation of (clockwise) with
UV fixed-point structure of the 3d Thirring model
UV fixed-point structure of the 3d Thirring model arxiv:1006.3747 [hep-th] (PRD accepted) Lukas Janssen in collaboration with Holger Gies Friedrich-Schiller-Universität Jena Theoretisch-Physikalisches
From the finite to the transfinite: Λµ-terms and streams
From the finite to the transfinite: Λµ-terms and streams WIR 2014 Fanny He f.he@bath.ac.uk Alexis Saurin alexis.saurin@pps.univ-paris-diderot.fr 12 July 2014 The Λµ-calculus Syntax of Λµ t ::= x λx.t (t)u
Large β 0 corrections to the energy levels and wave function at N 3 LO
Large β corrections to the energy levels and wave function at N LO Matthias Steinhauser, University of Karlsruhe LCWS5, March 5 [In collaboration with A. Penin and V. Smirnov] M. Steinhauser, LCWS5, March
PARTIAL NOTES for 6.1 Trigonometric Identities
PARTIAL NOTES for 6.1 Trigonometric Identities tanθ = sinθ cosθ cotθ = cosθ sinθ BASIC IDENTITIES cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ PYTHAGOREAN IDENTITIES sin θ + cos θ =1 tan θ +1= sec θ 1 + cot
Lifting Entry (continued)
ifting Entry (continued) Basic planar dynamics of motion, again Yet another equilibrium glide Hypersonic phugoid motion Planar state equations MARYAN 1 01 avid. Akin - All rights reserved http://spacecraft.ssl.umd.edu
AdS black disk model for small-x DIS
AdS black disk model for small-x DIS Miguel S. Costa Faculdade de Ciências da Universidade do Porto 0911.0043 [hep-th], 1001.1157 [hep-ph] Work with. Cornalba and J. Penedones Rencontres de Moriond, March
SPONTANEOUS GENERATION OF GEOMETRY IN 4D
SPONTANEOUS GENERATION OF GEOMETRY IN 4D Dani Puigdomènch Dual year Russia-Spain: Particle Physics, Nuclear Physics and Astroparticle Physics 10/11/11 Based on arxiv: 1004.3664 and wor on progress. by
Distances in Sierpiński Triangle Graphs
Distances in Sierpiński Triangle Graphs Sara Sabrina Zemljič joint work with Andreas M. Hinz June 18th 2015 Motivation Sierpiński triangle introduced by Wac law Sierpiński in 1915. S. S. Zemljič 1 Motivation
3+1 Splitting of the Generalized Harmonic Equations
3+1 Splitting of the Generalized Harmonic Equations David Brown North Carolina State University EGM June 2011 Numerical Relativity Interpret general relativity as an initial value problem: Split spacetime
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
ECE 308 SIGNALS AND SYSTEMS FALL 2017 Answers to selected problems on prior years examinations
ECE 308 SIGNALS AND SYSTEMS FALL 07 Answers to selected problems on prior years examinations Answers to problems on Midterm Examination #, Spring 009. x(t) = r(t + ) r(t ) u(t ) r(t ) + r(t 3) + u(t +
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
GegenbauerC3General. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation
GegenbauerC3General Notations Traditional name Gegenbauer function Traditional notation C Ν Λ z Mathematica StandardForm notation GegenbauerCΝ, Λ, z Primary definition 07.4.0.000.0 C Λ Ν z Λ Π Ν Λ F Ν,
ANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?
Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least
Srednicki Chapter 55
Srednicki Chapter 55 QFT Problems & Solutions A. George August 3, 03 Srednicki 55.. Use equations 55.3-55.0 and A i, A j ] = Π i, Π j ] = 0 (at equal times) to verify equations 55.-55.3. This is our third
MATH423 String Theory Solutions 4. = 0 τ = f(s). (1) dτ ds = dxµ dτ f (s) (2) dτ 2 [f (s)] 2 + dxµ. dτ f (s) (3)
![MATH423 String Theory Solutions 4. = 0 τ = f(s). (1) dτ ds = dxµ dτ f (s) (2) dτ 2 [f (s)] 2 + dxµ. dτ f (s) (3)](/thumbs/81/84712331.jpg "MATH423 String Theory Solutions 4. = 0 τ = f(s). (1) dτ ds = dxµ dτ f (s) (2) dτ 2 [f (s)] 2 + dxµ. dτ f (s) (3)")
1. MATH43 String Theory Solutions 4 x = 0 τ = fs). 1) = = f s) ) x = x [f s)] + f s) 3) equation of motion is x = 0 if an only if f s) = 0 i.e. fs) = As + B with A, B constants. i.e. allowe reparametrisations
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
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
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
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
Second Order Partial Differential Equations
Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y