Adachi-Tamura [4] [5] Gérard- Laba Adachi [1] 1
|
|
- Δεσποίνη Αλεξόπουλος
- 5 χρόνια πριν
- Προβολές:
Transcript
1 207 : msjmeeting-207sep-07i00 ( ) Abstract 989 Korotyaev Schrödinger Gérard Laba Multiparticle quantum scattering in constant magnetic fields - propagator ( ). ( ) 20 Sigal-Soffer [22] 987 Gérard- Laba [9] [0] Adachi-Tamura [4] [5] Gérard- Laba Adachi [] Skibsted [2] 0 2 Møller [8] Adachi [2] 0 2 Yajima [23] 3 Nakamura [20] Møller-Skibsted [9] (Mourre ) Nakamura Korotyaev [5] Mathematics Subject Classification: 8Q05. Keywords: Magnetic fields, scattering theory, time-periodic quantum system. mkawa@rs.tus.ac.jp
2 2. R 2 B(t) = (0, 0, B(t)) x = (x, x 2 ) R 2, p = i = ( i, i 2 ) m > 0, q R\{0} H 0 (t) H 0 (t) = 2m (p qa(t, x))2, A(t, x) = B(t) 2 ( x 2, x ). L 2 (R 2 ) A(t, x) V H(t) = H 0 (t) + V V V (x) V (x) (V) : (V) V C(R 2 ) σ > 0 C > 0 V (x) C( + x ) σ H 0 (t) propagator U 0 (t, 0), H(t) propagator U(t, 0) U(t, 0) U 0 (t, 0) B(t) B(t) { B t I B, B(t) = I B := [nt, nt + T B ), I 0 := + T B, (n + )T ), 0 t I 0, n Z n Z[nT B > 0, 0 < T B < T t I B B = (0, 0, B) ( on ) t I 0 ( off ) B(t + T ) = B(t), t on, off H B, Schrödinger H 0 H B := 2m (p qa(x))2, A(x) = B 2 ( x 2, x ), H 0 := p2 2m H 0 (t) H 0 (t) = { H B t I B, H 0 t I 0
3 U 0 (t, 0) { e i(t nt )H B U 0 (nt, 0) t [nt, nt + T B ), U 0 (t, 0) = e i(t nt T B)H 0 e it BH B U 0 (nt, 0) t [nt + T B, (n + )T ). off T 0 := T T B U 0 (T, 0) = e it 0H 0 e it BH B, U 0 (nt, 0) = (U 0 (T, 0)) n U 0 (t, 0) U 0 (T, 0) ω = qb m, ω = ω 2, ω = ω 2 = ω 4. ω Larmor Lemma 2.. t 0 S 0 0(t; x) S B 0 (t; x, y) S0(t; 0 x) := /(2t) 2πit eix2, S0 B m ω (t; x, y) := eim ω cot( ω t)x2 /2 e im ω ( ˆR( ωt)x) y/ sin( ω t) e im ω cot( ω t)y2 /2 2πi sin( ω t) e ith 0 ψ = S0(t; 0 x y)ψ(y)dy, R 2 e ith B ψ = S0 B (t; x, y)ψ(y)dy R 2 ( cos θ sin θ ˆR(θ) = sin θ cos θ H B Avron-Herbst-Simon [6], Adachi-Kawamoto [3] U 0 (T, 0) S 0 (T ; x, y) S 0 (T ; x, y) = 2πic θ e ix2 /(2θ ) e i( ˆR(ϕ )x) y/(c θ ) e iσ y2 /(2θ ), c, σ, θ, ϕ (2) L, L 2, L 2, L 22 θ = L 2 L 22, c = L 22, σ = L L 22, ϕ = ωt B. )
4 U 0 (nt, 0) = (U 0 (T, 0)) n S 0 (nt ; x, y) S 0 (nt ; x, y) = 2πic n θ n e ix2 /(2θ n) e i( ˆR(ϕ n)x) y/(c nθ n) e iσny2 /(2θ n). () c n, σ n θ n, ϕ n ( = θ n+ c 2 σ = c n+ θ n+ ( σ n+ = σ n θ n+ c 2 n ϕ n+ = ϕ + ϕ n. ) θ + c σ c n (θ /σ + θ n ), ) + θ n (c σ ) 2 (θ /σ + θ n ), c 2 n(θ /σ + θ n ), ϕ n, θ n ϕ n = n ωt B θ n θ n θ n+ = L 2θ n + L 22 L θ n + L 2, L = cos( ω T B ) ω T 0 sin( ω T B ), L 2 = m ω ( ω T 0 cos( ω T B ) + sin( ω T B )), (2) L 2 = m ω sin( ω T B ), L 22 = cos( ω T B ) α ± α = L α + L 2 L 2 α + L 22 D D = λ 2 0, λ 0 := L + L 22 2 = cos( ω T B ) 2 ω T 0 sin( ω T B ), λ ± := λ 0 ± λ 2 0, D > 0 µ n := λn + λ n λ + λ = λn + λ n 2 λ 2 0 (3) L 2 µ n θ n = L 22 µ n µ n
5 D off : T 0,cr := cos( ωt B ) ω sin( ωt B ), T 0 T 0,cr λ 0 + 0, i.e. D = λ 2 0 0, 0 < T 0 < T 0,cr (λ 0 + )(λ 0 ) < 0, i.e. D = λ 2 0 < 0. T 0,cr Theorem 2.2 (Adachi- K). propagator U 0 (nt, 0) () T 0 T 0,cr θ n = L 2 µ n L 22 µ n µ n, c n = L 22 µ n µ n, σ n = L µ n µ n L 22 µ n µ n, ϕ n = n ωt B T 0 = T 0,cr θ n = nl 2 nl 22 (n )λ 0, σ n = nl (n )λ 0 nl 22 (n )λ 0, c n = λ n 0 {nl 22 (n )λ 0 }, ϕ n = n ωt B Remark 2.3. [3] t R U 0 (t, 0) t nt L L U 0 (nt, 0)ψ L (R 2 ) 2πc nθ n ψ L (R 2 ) (4) c n θ n = { L 2 µ n T 0 T 0,cr, L 2 nλ n 0 T 0 = T 0,cr T 0 < T 0,cr λ 2 0 < (3) λ ± C λ ± < (4) nt e i(nt )H B L L U 0 (T, 0) ([3]) T 0 = T 0,cr L L (4) n e i(nt )H 0 off T 0,cr off T 0,cr
6 T 0 > T 0,cr λ 0 < λ > λ + < L 2 0 n T 0 T 0,res := sin( ω T B) ω cos( ω T B ) U 0 (nt, 0)ψ L (R 2 ) 2π µ n L 2 ψ L (R 2 ) Ce νn ψ L (R 2 ), ν > 0 Proposition 2.4. T 0 > T 0,cr T 0 T 0,res θ > 0, ψ L 2 (R 2 ) C > 0 ( + x ) θ U 0 (t, 0)( + x ) θ ψ dt C ψ L 2 (R 2 ) L 2 (R 2 ) R Remark 2.3 t R t t n = nt ( + x ) θ U 0 (nt, 0)( + x ) θ ψ C ψ L 2 (R 2 ) L 2 (R 2 ) (5) n Z Riesz-Thorin p 2, q = p/(p ) U 0 (nt, 0)ψ L p (R 2 ) Ce 2νn(/2 /p) ψ L q (R 2 ). Kato [4] ( + x ) θ U 0 (nt, 0)( + x ) θ ψ L 2 (R 2 ) Ce 2νn/p ( + x ) θ 2 L p (R 2 ) ψ L 2 (R 2 ) p 2 θ > 0 pθ > 2 p (5) T 0 > T 0,cr [8], [], [2], [7], [23] K := L 2 (T; L 2 (R 2 )), T = R/T Z ψ K, ψ(t) = ψ(t; ) L σ 0, L σ (L σ 0 ψ)(t) = U 0 (t, t σ) ψ(t σ), (L σ ψ)(t) = U(t, t σ) ψ(t σ).
7 L σ 0, L σ K Stone K Ĥ0, Ĥ e iσĥ0 ψ = L σ 0 ψ, e iσĥ ψ = L σ ψ Ĥ0 U 0 (t, 0) Floquet Floquet : Lemma 2.5. [ Yajima [24]] f K Ĥf = λf f(t) L 2 (R 2 )- f(t) = e iλt U(t, 0)f(0), U(T, 0)f(0) = e iλt f(0) U(T, 0)φ = e it λ φ f(t) := e itλ U(t, 0)φ D(Ĥ) Ĥf = λf. Lemma 2.6 (Kitada-Yajima [7], Enss-Veselić [8]). L 2 (R 2 ) = L 2 c(u(t, 0)) L 2 p(u(t, 0)) L 2 c(u(t, 0)) L 2 p(u(t, 0)) U(T, 0) Lemma 2.5 Floquet Ĥ Floquet Howland-Yajima method (Howland [] [2], Yajima [23]) Lemma 2.7 (Howland [] [2], Yajima [23]). W ± iσĥ0 := s lim eiσĥe σ ± Ran(W ± ) = K ac (Ĥ) Kac(Ĥ) K Ĥ W ± := s lim t ± U(t, 0) U 0 (t, 0) Ran(W ± ) = L 2 ac(u(t, 0)) L 2 ac(u(t, 0)) L 2 (R 2 ) U(T, 0) Theorem 2.8 (Adachi- K [3]). V (V) on, off T B, T 0 (T 0) 0 < ω T B < π, (T ) T 0 > T 0,cr, (T 2) T 0 T 0,res, if π/2 < ω T B < π Ran ( W ±) = L 2 ac(u(t, 0))
8 3. Howland-Yajima method Kato s smooth perturbation method (Kato [4])., i.e., V (Ĥ + i) K 2., i.e., ρ = V /2 sign(v ), ρ 2 = V /2, ψ K Γ R, sup λ Γ, ±µ>0 ρ (Ĥ λ iµ) ρ 2 ψ K C ψ K 3. W ± 3 Cook-Kuroda method W ± ψ C 0 (R 2 ) 0 V U 0 (t, 0)ψ L 2 (R 2 ) dt C 2.4 θ = σ 0 C C V U 0 (t, 0)ψ L 2 (R 2 ) dt 0 ( + x ) σ U 0 (t, 0)( + x ) σ ( + B(L 2 (R 2 )) x )σ ψ L 2 (R 2 ) dt σ > 0 2. Ĥ0 Kato-Kuroda [6] Ĥ H 0 Imz > 0 Floquet ( Imz < 0 ) ρ (Ĥ0 z) ρ 2 ψ = i { T n= 0 t + 0 e i(t+nt s)z ρ U 0 (t + nt, s)(ρ 2 ψ)(s)ds e i(t s)z ρ U 0 (t, s)(ρ 2 ψ)(s)ds } 2.4. U 0 (t, 0)
9 4. Remarks Remark 4.. Hill f (t) + ( ) 2 qb(t) f(t) = 0 2m Remark 4.2. Bony-Carles-Häfner-Michel [7] V (x) C(log( + x )) ε Ishida [3] log Remark 4.3. T 0 = T 0,cr 3 [5] 2 [5], [3] References [] Adachi, T.: On spectral and scattering theory for N-body Schrödinger operators in a constant magnetic field, Rev. in Math. Phys., 2, (2002) [2] Adachi, T.: Asymptotic completeness for N-body quantum systems with longrange interactions in a time-periodic electric field, Comm. Math. Phys., 275, (2007) [3] Adachi, T., Kawamoto, M.: Quantum scattering in a periodically pulsed magnetic field, A. H. P., 7, , (206) [4] Adachi, T., Tamura, H.: Asymptotic completeness for long-range many particle systems with Stark effect, J. Math. Sci. Univ. Tokyo, 2, 77-6 (995) [5] Adachi, T., Tamura, H.: Asymptotic completeness for long-range many particle systems with Stark effect, II, Comm. Math. Phys. 74, (996) [6] Avron, J.E., Herbst, I.W., Simon, B.: Schrödinger operators with magnetic fields. I. General interactions. Duke Math. J. 45, (978) [7] Bony, J. F., Carles, R., Häfner, D., Michel, L.: Scattering theory for the Schrödinger equation with repulsive potential, J. Math. Pures Appl. 84, (2005) [8] Enss, V., Veselić, K.: Bound states and propagtiong states for time-dependent hamiltonians, Ann. de l I. H. P., sect. A 39, 59-9 (983) [9] Gérard, C., Laba, I.: Scattering theory for N-particle systems in constant magnetic fields, Duke Math. J., 76, (994) [0] Gérard, C., Laba, I.: Scattering theory for N-particle systems in constant magnetic fields, II, Comm. in P.D.E., 20, (996) [] Howland, J.S.: Scattering theory for time-dependent Hamiltonians. Math. Ann. 207, (974)
10 [2] Howland, J.S.: Scattering theory for Hamiltonians periodic in time. Indiana Univ. Math. J. 28, (979) [3] Ishida, A.: The borderline of the short-range condition for the repulsive Hamiltonian, J. Math. Anal. and Appl., 438, (206) [4] Kato, T.: Wave operators and similarity for some non-selfadjoint operators. Math. Ann. 62, (966) [5] Korotyaev, E. L.: On scattering in an external, homogeneous, time-periodic magnetic field, Math. USSR-Sb. 66, (990) [6] Kato, T., Kuroda, S. T.: The abstract theory of scattering. Rocky Mt. J. Math., 27-7 (97) [7] Kitada, H., Yajima, K.: Bound states and scattering states for time periodic hamiltonians, Ann. de l I. H. P., sect. A 39, (983) [8] Møller, J. S.: Two-body short-range systems in a time-periodic electric field, Duke Math. J., 05, (2000) [9] Møller, J. S., Skibsted, E.: Spectral theory of time-periodic many-body systems, Adv. in Math., 88, (2004) [20] Nakamura, S.: Asymptotic completeness for three-body Schrödinger equations with time-periodic potentials, J. Fac. Sci. Univ. Tokyo. Sect. IA, Math., 33, (986) [2] Skibsted, E.: Asymptotic completeness for particles in combined constant electric and magnetic fields, II, Duke Math. J., 89, (997) [22] Sigal, I. M., Soffer, A.: The N-particle scattering problem: asymptotic completeness for short-range quantum system, Ann. of Math. 25, (987) [23] Yajima, K.: Schrödinger equations with potentials periodic in time. J. Math. Soc. Jpn. 87, (977) [24] Yajima, K.: Resonances for the AC-Stark effect, Comm. Math. Phys., 87, (982)
Solutions - Chapter 4
Solutions - Chapter Kevin S. Huang Problem.1 Unitary: Ût = 1 ī hĥt Û tût = 1 Neglect t term: 1 + hĥ ī t 1 īhĥt = 1 + hĥ ī t ī hĥt = 1 Ĥ = Ĥ Problem. Ût = lim 1 ī ] n hĥ1t 1 ī ] hĥt... 1 ī ] hĥnt 1 ī ]
Διαβάστε περισσότερα1. 3. ([12], Matsumura[13], Kikuchi[10] ) [12], [13], [10] ( [12], [13], [10]
3. 3 2 2) [2] ) ) Newton[4] Colton-Kress[2] ) ) OK) [5] [] ) [2] Matsumura[3] Kikuchi[] ) [2] [3] [] 2 ) 3 2 P P )+ P + ) V + + P H + ) [2] [3] [] P V P ) ) V H ) P V ) ) ) 2 C) 25473) 2 3 Dermenian-Guillot[3]
Διαβάστε περισσότεραPhys460.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
Διαβάστε περισσότεραu = g(u) in R N, u > 0 in R N, u H 1 (R N ).. (1), u 2 dx G(u) dx : H 1 (R N ) R
2017 : msjmeeting-2017sep-05i002 ( ) 1.. u = g(u) in R N, u > 0 in R N, u H 1 (R N ). (1), N 2, g C 1 g(0) = 0. g(s) = s + s p. (1), [8, 9, 17],., [15] g. (1), E(u) := 1 u 2 dx G(u) dx : H 1 (R N ) R 2
Διαβάστε περισσότεραJ. of Math. (PRC) Banach, , X = N(T ) R(T + ), Y = R(T ) N(T + ). Vol. 37 ( 2017 ) No. 5
Vol. 37 ( 2017 ) No. 5 J. of Math. (PRC) 1,2, 1, 1 (1., 225002) (2., 225009) :. I +AT +, T + = T + (I +AT + ) 1, T +. Banach Hilbert Moore-Penrose.. : ; ; Moore-Penrose ; ; MR(2010) : 47L05; 46A32 : O177.2
Διαβάστε περισσότεραDispersive estimates for rotating fluids and stably stratified fluids
特別講演 17 : msjmeeting-17sep-5i4 Dispersive estimates for rotating fluids and stably stratified fluids ( ) 1. Navier-Stokes (1.1) Boussinesq (1.) t v + (v )v = v q t >, x R, v = t >, x R, t v + (v )v = v
Διαβάστε περισσότεραJ. of Math. (PRC) 6 n (nt ) + n V = 0, (1.1) n t + div. div(n T ) = n τ (T L(x) T ), (1.2) n)xx (nt ) x + nv x = J 0, (1.4) n. 6 n
Vol. 35 ( 215 ) No. 5 J. of Math. (PRC) a, b, a ( a. ; b., 4515) :., [3]. : ; ; MR(21) : 35Q4 : O175. : A : 255-7797(215)5-15-7 1 [1] : [ ( ) ] ε 2 n n t + div 6 n (nt ) + n V =, (1.1) n div(n T ) = n
Διαβάστε περισσότεραMetastable states in a model of spin dependent point interactions. claudio cacciapuoti, raffaele carlone, rodolfo figari
Metastable states in a model of spin dependent point interactions claudio cacciapuoti, raffaele carlone, rodolfo figari Metastable states in a model of spin dependent point interactions Metastable states
Διαβάστε περισσότερα([28] Bao-Feng Feng (UTP-TX), ( ), [20], [16], [24]. 1 ([3], [17]) p t = 1 2 κ2 T + κ s N -259-
5,..,. [8]..,,.,.., Bao-Feng Feng UTP-TX,, UTP-TX,,. [0], [6], [4].. ps ps, t. t ps, 0 = ps. s 970 [0] []. [3], [7] p t = κ T + κ s N -59- , κs, t κ t + 3 κ κ s + κ sss = 0. T s, t, Ns, t., - mkdv. mkdv.
Διαβάστε περισσότεραPrey-Taxis Holling-Tanner
Vol. 28 ( 2018 ) No. 1 J. of Math. (PRC) Prey-Taxis Holling-Tanner, (, 730070) : prey-taxis Holling-Tanner.,,.. : Holling-Tanner ; prey-taxis; ; MR(2010) : 35B32; 35B36 : O175.26 : A : 0255-7797(2018)01-0140-07
Διαβάστε περισσότεραOn the Galois Group of Linear Difference-Differential Equations
On the Galois Group of Linear Difference-Differential Equations Ruyong Feng KLMM, Chinese Academy of Sciences, China Ruyong Feng (KLMM, CAS) Galois Group 1 / 19 Contents 1 Basic Notations and Concepts
Διαβάστε περισσότεραChapter 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
Διαβάστε περισσότεραHeisenberg 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,
Διαβάστε περισσότεραChapter 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
Διαβάστε περισσότεραMarkov chains model reduction
Markov chains model reduction C. Landim Seminar on Stochastic Processes 216 Department of Mathematics University of Maryland, College Park, MD C. Landim Markov chains model reduction March 17, 216 1 /
Διαβάστε περισσότεραA General Note on δ-quasi Monotone and Increasing Sequence
International Mathematical Forum, 4, 2009, no. 3, 143-149 A General Note on δ-quasi Monotone and Increasing Sequence Santosh Kr. Saxena H. N. 419, Jawaharpuri, Badaun, U.P., India Presently working in
Διαβάστε περισσότεραSingle-value extension property for anti-diagonal operator matrices and their square
1 215 1 Journal of East China Normal University Natural Science No. 1 Jan. 215 : 1-56412151-95-8,, 71119 :, Hilbert. : ; ; : O177.2 : A DOI: 1.3969/j.issn.1-5641.215.1.11 Single-value extension property
Διαβάστε περισσότεραD Alembert s Solution to the Wave Equation
D Alembert s Solution to the Wave Equation MATH 467 Partial Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Objectives In this lesson we will learn: a change of variable technique
Διαβάστε περισσότεραHomomorphism in Intuitionistic Fuzzy Automata
International Journal of Fuzzy Mathematics Systems. ISSN 2248-9940 Volume 3, Number 1 (2013), pp. 39-45 Research India Publications http://www.ripublication.com/ijfms.htm Homomorphism in Intuitionistic
Διαβάστε περισσότεραΠ Ο Λ Ι Τ Ι Κ Α Κ Α Ι Σ Τ Ρ Α Τ Ι Ω Τ Ι Κ Α Γ Ε Γ Ο Ν Ο Τ Α
Α Ρ Χ Α Ι Α Ι Σ Τ Ο Ρ Ι Α Π Ο Λ Ι Τ Ι Κ Α Κ Α Ι Σ Τ Ρ Α Τ Ι Ω Τ Ι Κ Α Γ Ε Γ Ο Ν Ο Τ Α Σ η µ ε ί ω σ η : σ υ ν ά δ ε λ φ ο ι, ν α µ ο υ σ υ γ χ ω ρ ή σ ε τ ε τ ο γ ρ ή γ ο ρ ο κ α ι α τ η µ έ λ η τ ο ύ
Διαβάστε περισσότερα11 Drinfeld. k( ) = A/( ) A K. [Hat1, Hat2] k M > 0. Γ 1 (M) = γ SL 2 (Z) f : H C. ( ) az + b = (cz + d) k f(z) ( z H, γ = cz + d Γ 1 (M))
Drinfeld Drinfeld 29 8 8 11 Drinfeld [Hat3] 1 p q > 1 p A = F q [t] A \ F q d > 0 K A ( ) k( ) = A/( ) A K Laurent F q ((1/t)) 1/t C Drinfeld Drinfeld p p p [Hat1, Hat2] 1.1 p 1.1.1 k M > 0 { Γ 1 (M) =
Διαβάστε περισσότεραGeodesic paths for quantum many-body systems
Geodesic paths for quantum many-body systems Michael Tomka, Tiago Souza, Steve Rosenberg, and Anatoli Polkovnikov Department of Physics Boston University Group: Condensed Matter Theory June 6, 2016 Workshop:
Διαβάστε περισσότεραHomework 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
Διαβάστε περισσότεραDifferential equations
Differential equations Differential equations: An equation inoling one dependent ariable and its deriaties w. r. t one or more independent ariables is called a differential equation. Order of differential
Διαβάστε περισσότεραVol. 37 ( 2017 ) No. 3. J. of Math. (PRC) : A : (2017) k=1. ,, f. f + u = f φ, x 1. x n : ( ).
Vol. 37 ( 2017 ) No. 3 J. of Math. (PRC) R N - R N - 1, 2 (1., 100029) (2., 430072) : R N., R N, R N -. : ; ; R N ; MR(2010) : 58K40 : O192 : A : 0255-7797(2017)03-0467-07 1. [6], Mather f : (R n, 0) R
Διαβάστε περισσότεραCHAPTER 101 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD
CHAPTER FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD EXERCISE 36 Page 66. Determine the Fourier series for the periodic function: f(x), when x +, when x which is periodic outside this rge of period.
Διαβάστε περισσότεραLecture 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
Διαβάστε περισσότεραANSWERSHEET (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
Διαβάστε περισσότεραSPECIAL 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
Διαβάστε περισσότεραLow Frequency Plasma Conductivity in the Average-Atom Approximation
Low Frequency Plasma Conductivity in the Average-Atom Approximation Walter Johnson & Michael Kuchiev Physical Review E 78, 026401 (2008) 1. Review of Average-Atom Linear Response Theory 2. Demonstration
Διαβάστε περισσότεραLifting 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
Διαβάστε περισσότεραEquations. BSU Math 275 sec 002,003 Fall 2018 (Ultman) Final Exam Notes 1. du dv. FTLI : f (B) f (A) = f dr. F dr = Green s Theorem : y da
BSU Math 275 sec 002,003 Fall 2018 (Ultman) Final Exam Notes 1 Equations r(t) = x(t) î + y(t) ĵ + z(t) k r = r (t) t s = r = r (t) t r(u, v) = x(u, v) î + y(u, v) ĵ + z(u, v) k S = ( ( ) r r u r v = u
Διαβάστε περισσότεραME 365: SYSTEMS, MEASUREMENTS, AND CONTROL (SMAC) I
ME 365: SYSTEMS, MEASUREMENTS, AND CONTROL SMAC) I Dynamicresponseof 2 nd ordersystem Prof.SongZhangMEG088) Solutions to ODEs Forann@thorderLTIsystem a n yn) + a n 1 y n 1) ++ a 1 "y + a 0 y = b m u m)
Διαβάστε περισσότεραHigher 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)
Διαβάστε περισσότεραDurbin-Levinson recursive method
Durbin-Levinson recursive method A recursive method for computing ϕ n is useful because it avoids inverting large matrices; when new data are acquired, one can update predictions, instead of starting again
Διαβάστε περισσότεραDynamics of cold molecules in external electromagnetic fields. Roman Krems University of British Columbia
Dynamics of cold molecules in external electromagnetic fields Roman Krems University of British Columbia UBC group: Zhiying Li Timur Tscherbul Erik Abrahamsson Sergey Alyabishev Chris Hemming Collaborations:
Διαβάστε περισσότεραGeneral 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,
Διαβάστε περισσότεραΑπόκριση 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
Διαβάστε περισσότεραExercises 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
Διαβάστε περισσότεραDerivation of Optical-Bloch Equations
Appendix C Derivation of Optical-Bloch Equations In this appendix the optical-bloch equations that give the populations and coherences for an idealized three-level Λ system, Fig. 3. on page 47, will be
Διαβάστε περισσότεραThe Negative Neumann Eigenvalues of Second Order Differential Equation with Two Turning Points
Applied Mathematical Sciences, Vol. 3, 009, no., 6-66 The Negative Neumann Eigenvalues of Second Order Differential Equation with Two Turning Points A. Neamaty and E. A. Sazgar Department of Mathematics,
Διαβάστε περισσότεραTakeaki 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
Διαβάστε περισσότεραInflation and Reheating in Spontaneously Generated Gravity
Univesità di Bologna Inflation and Reheating in Spontaneously Geneated Gavity (A. Ceioni, F. Finelli, A. Tonconi, G. Ventui) Phys.Rev.D81:123505,2010 Motivations Inflation (FTV Phys.Lett.B681:383-386,2009)
Διαβάστε περισσότεραd dx x 2 = 2x d dx x 3 = 3x 2 d dx x n = nx n 1
d dx x 2 = 2x d dx x 3 = 3x 2 d dx x n = nx n1 x dx = 1 2 b2 1 2 a2 a b b x 2 dx = 1 a 3 b3 1 3 a3 b x n dx = 1 a n +1 bn +1 1 n +1 an +1 d dx d dx f (x) = 0 f (ax) = a f (ax) lim d dx f (ax) = lim 0 =
Διαβάστε περισσότεραŁs t r t rs tø r P r s tø PrØ rø rs tø P r s r t t r s t Ø t q s P r s tr. 2stŁ s q t q s t rt r s t s t ss s Ø r s t r t. Łs t r t t Ø t q s
Łs t r t rs tø r P r s tø PrØ rø rs tø P r s r t t r s t Ø t q s P r s tr st t t t Ø t q s ss P r s P 2stŁ s q t q s t rt r s t s t ss s Ø r s t r t P r røs r Łs t r t t Ø t q s r Ø r t t r t q t rs tø
Διαβάστε περισσότεραACTA 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
Διαβάστε περισσότεραSCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018
Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals
Διαβάστε περισσότεραBessel functions. ν + 1 ; 1 = 0 for k = 0, 1, 2,..., n 1. Γ( n + k + 1) = ( 1) n J n (z). Γ(n + k + 1) k!
Bessel functions The Bessel function J ν (z of the first kind of order ν is defined by J ν (z ( (z/ν ν Γ(ν + F ν + ; z 4 ( k k ( Γ(ν + k + k! For ν this is a solution of the Bessel differential equation
Διαβάστε περισσότεραUniform 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
Διαβάστε περισσότεραThe k-α-exponential Function
Int Journal of Math Analysis, Vol 7, 213, no 11, 535-542 The --Exponential Function Luciano L Luque and Rubén A Cerutti Faculty of Exact Sciences National University of Nordeste Av Libertad 554 34 Corrientes,
Διαβάστε περισσότεραA Two-Sided Laplace Inversion Algorithm with Computable Error Bounds and Its Applications in Financial Engineering
Electronic Companion A Two-Sie Laplace Inversion Algorithm with Computable Error Bouns an Its Applications in Financial Engineering Ning Cai, S. G. Kou, Zongjian Liu HKUST an Columbia University Appenix
Διαβάστε περισσότεραReminders: 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
Διαβάστε περισσότεραΚβαντομηχανική Ι Λύσεις προόδου. Άσκηση 1
Κβαντομηχανική Ι Λύσεις προόδου Άσκηση 1 ψ(x) = A Sin (k x), < x < α) Sin (k x) = eikx e ikx i Mε πιθανές τιμές ορμής p = ± ħk, από τον τύπο του De Broglie. Kαθεμιά έχει πιθανότητα 50%. b) p = ψ p ψ =
Διαβάστε περισσότεραCoefficient Inequalities for a New Subclass of K-uniformly Convex Functions
International Journal of Computational Science and Mathematics. ISSN 0974-89 Volume, Number (00), pp. 67--75 International Research Publication House http://www.irphouse.com Coefficient Inequalities for
Διαβάστε περισσότεραProblem 7.19 Ignoring reflection at the air soil boundary, if the amplitude of a 3-GHz incident wave is 10 V/m at the surface of a wet soil medium, at what depth will it be down to 1 mv/m? Wet soil is
Διαβάστε περισσότεραJournal of East China Normal University (Natural Science) DGH. Stability of peakons for the DGH equation. CHEN Hui-ping
5 2010 9 ) Journal of East China Normal University Natural Science) No. 5 Sep. 2010 : 1000-56412010)05-0067-06 DGH, 226007) :,. DGH H 1.,,. : ; DGH ; : O29 : A Stability of peakons for the DGH equation
Διαβάστε περισσότεραSTABILITY FOR RAYLEIGH-BENARD CONVECTIVE SOLUTIONS OF THE BOLTZMANN EQUATION
STABILITY FOR RAYLEIGH-BENARD CONVECTIVE SOLUTIONS OF THE BOLTZMANN EQUATION L.Arkeryd, Chalmers, Goteborg, Sweden, R.Esposito, University of L Aquila, Italy, R.Marra, University of Rome, Italy, A.Nouri,
Διαβάστε περισσότεραPOSITIVE SOLUTIONS FOR A FUNCTIONAL DELAY SECOND-ORDER THREE-POINT BOUNDARY-VALUE PROBLEM
Electronic Journal of Differential Equations, Vol. 26(26, No. 4, pp.. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp POSITIVE SOLUTIONS
Διαβάστε περισσότεραSolutions to the Schrodinger equation atomic orbitals. Ψ 1 s Ψ 2 s Ψ 2 px Ψ 2 py Ψ 2 pz
Solutions to the Schrodinger equation atomic orbitals Ψ 1 s Ψ 2 s Ψ 2 px Ψ 2 py Ψ 2 pz ybridization Valence Bond Approach to bonding sp 3 (Ψ 2 s + Ψ 2 px + Ψ 2 py + Ψ 2 pz) sp 2 (Ψ 2 s + Ψ 2 px + Ψ 2 py)
Διαβάστε περισσότεραM a t h e m a t i c a B a l k a n i c a. On Some Generalizations of Classical Integral Transforms. Nina Virchenko
M a t h e m a t i c a B a l k a n i c a New Series Vol. 26, 212, Fasc. 1-2 On Some Generalizations of Classical Integral Transforms Nina Virchenko Presented at 6 th International Conference TMSF 211 Using
Διαβάστε περισσότεραBroadband Spatiotemporal Differential-Operator Representations For Velocity-Dependent Scattering
Broadband Spatiotemporal Differential-Operator Representations For Velocity-Dependent Scattering Dan Censor Ben Gurion University of the Negev Department of Electrical and Computer Engineering Beer Sheva,
Διαβάστε περισσότεραŒ ˆ Œ Ÿ Œˆ Ÿ ˆŸŒˆ Œˆ Ÿ ˆ œ, Ä ÞŒ Å Š ˆ ˆ Œ Œ ˆˆ
ˆ ˆŠ Œ ˆ ˆ Œ ƒ Ÿ 018.. 49.. 4.. 907Ä917 Œ ˆ Œ Ÿ Œˆ Ÿ ˆŸŒˆ Œˆ Ÿ ˆ œ, Ä ÞŒ Å Š ˆ ˆ Œ Œ ˆˆ.. ³μ, ˆ. ˆ. Ë μ μ,.. ³ ʲ μ ± Ë ²Ó Ò Ö Ò Í É Å μ ± ÊÎ μ- ² μ É ²Ó ± É ÉÊÉ Ô± ³ É ²Ó μ Ë ±, μ, μ Ö μ ² Ìμ μé Ê Ö ±
Διαβάστε περισσότεραForced Pendulum Numerical approach
Numerical approach UiO April 8, 2014 Physical problem and equation We have a pendulum of length l, with mass m. The pendulum is subject to gravitation as well as both a forcing and linear resistance force.
Διαβάστε περισσότεραJ. of Math. (PRC) Shannon-McMillan, , McMillan [2] Breiman [3] , Algoet Cover [10] AEP. P (X n m = x n m) = p m,n (x n m) > 0, x i X, 0 m i n. (1.
Vol. 35 ( 205 ) No. 4 J. of Math. (PRC), (, 243002) : a.s. Marov Borel-Catelli. : Marov ; Borel-Catelli ; ; ; MR(200) : 60F5 : O2.4; O236 : A : 0255-7797(205)04-0969-08 Shao-McMilla,. Shao 948 [],, McMilla
Διαβάστε περισσότεραg-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
Διαβάστε περισσότεραLTI Systems (1A) Young Won Lim 3/21/15
LTI Systems (1A) Copyright (c) 214 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version
Διαβάστε περισσότεραΑ Ρ Ι Θ Μ Ο Σ : 6.913
Α Ρ Ι Θ Μ Ο Σ : 6.913 ΠΡΑΞΗ ΚΑΤΑΘΕΣΗΣ ΟΡΩΝ ΔΙΑΓΩΝΙΣΜΟΥ Σ τ η ν Π ά τ ρ α σ ή μ ε ρ α σ τ ι ς δ ε κ α τ έ σ σ ε ρ ι ς ( 1 4 ) τ ο υ μ ή ν α Ο κ τ ω β ρ ί ο υ, η μ έ ρ α Τ ε τ ά ρ τ η, τ ο υ έ τ ο υ ς δ
Διαβάστε περισσότεραDissertation for the degree philosophiae doctor (PhD) at the University of Bergen
Dissertation for the degree philosophiae doctor (PhD) at the University of Bergen Dissertation date: GF F GF F SLE GF F D Ĉ = C { } Ĉ \ D D D = {z : z < 1} f : D D D D = D D, D = D D f f : D D
Διαβάστε περισσότερα第 61 回トポロジーシンポジウム講演集 2014 年 7 月於東北大学 Kontsevich-Kuperberg-Thurston ( ) Kontsevich-Kuperberg-Thurston Kontsevich Chern-Simons E. Witten Chern-
Kontsevich-Kuperberg-Thurston ( ) Kontsevich-Kuperberg-Thurston Kontsevich Chern-Simons 3 1 1989 E. Witten Chern-Simons 3 ( ) ([14]) Witten 3 Chern-Simons M. Kontsevich [5], S. Axerod I. M. Singer [2]
Διαβάστε περισσότερα2.019 Design of Ocean Systems. Lecture 6. Seakeeping (II) February 21, 2011
2.019 Design of Ocean Systems Lecture 6 Seakeeping (II) February 21, 2011 ω, λ,v p,v g Wave adiation Problem z ζ 3 (t) = ζ 3 cos(ωt) ζ 3 (t) = ω ζ 3 sin(ωt) ζ 3 (t) = ω 2 ζ3 cos(ωt) x 2a ~n Total: P (t)
Διαβάστε περισσότερα[I2], [IK1], [IK2], [AI], [AIK], [INA], [IN], [IK2], [IA1], [I3], [IKP], [BIK], [IA2], [KB]
(Akihiko Inoue) Graduate School of Science, Hiroshima University (Yukio Kasahara) Graduate School of Science, Hokkaido University Mohsen Pourahmadi, Department of Statistics, Texas A&M University 1, =
Διαβάστε περισσότεραx xn w(x) = 0 ( n N)
( ). Wierstrass Bernstein ([]) lim x xn w(x) = 0 ( n N) R w fw C 0 (R) lim (f P n)w L n (R) = 0 {P n } w Bernstein 950 ([5], [8] ) 970 Freud Freud weights w α (x) = exp( x α ) ([3] ) α Bernstein w α Christoffel
Διαβάστε περισσότερα172,,,,. P,. Box (1980)P, Guttman (1967)Rubin (1984)P, Meng (1994), Gelman(1996)De la HorraRodriguez-Bernal (2003). BayarriBerger (2000)P P.. : Casell
20104 Chinese Journal of Applied Probability and Statistics Vol.26 No.2 Apr. 2010 P (,, 200083) P P. Wang (2006)P, P, P,. : P,,,. : O212.1, O212.8. 1., (). : X 1, X 2,, X n N(θ, σ 2 ), σ 2. H 0 : θ = θ
Διαβάστε περισσότεραSOLUTIONS 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
Διαβάστε περισσότεραSpherical Coordinates
Spherical Coordinates MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Spherical Coordinates Another means of locating points in three-dimensional space is known as the spherical
Διαβάστε περισσότεραGraded 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
Διαβάστε περισσότεραd 2 y dt 2 xdy dt + d2 x
y t t ysin y d y + d y y t z + y ty yz yz t z y + t + y + y + t y + t + y + + 4 y 4 + t t + 5 t Ae cos + Be sin 5t + 7 5 y + t / m_nadjafikhah@iustacir http://webpagesiustacir/m_nadjafikhah/courses/ode/fa5pdf
Διαβάστε περισσότεραGlobal nonlinear stability of steady solutions of the 3-D incompressible Euler equations with helical symmetry and with no swirl
Around Vortices: from Cont. to Quantum Mech. Global nonlinear stability of steady solutions of the 3-D incompressible Euler equations with helical symmetry and with no swirl Maicon José Benvenutti (UNICAMP)
Διαβάστε περισσότεραSOLUTIONS 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
Διαβάστε περισσότεραMolekulare Ebene (biochemische Messungen) Zelluläre Ebene (Elektrophysiologie, Imaging-Verfahren) Netzwerk Ebene (Multielektrodensysteme) Areale (MRT, EEG...) Gene Neuronen Synaptische Kopplung kleine
Διαβάστε περισσότεραMock Exam 7. 1 Hong Kong Educational Publishing Company. Section A 1. Reference: HKDSE Math M Q2 (a) (1 + kx) n 1M + 1A = (1) =
Mock Eam 7 Mock Eam 7 Section A. Reference: HKDSE Math M 0 Q (a) ( + k) n nn ( )( k) + nk ( ) + + nn ( ) k + nk + + + A nk... () nn ( ) k... () From (), k...() n Substituting () into (), nn ( ) n 76n 76n
Διαβάστε περισσότεραΦΥΣΙΚΗΣ ΣΤΟΙΧΕΙΩΔΩΝ ΣΩΜΑΤΙΔΙΩΝ ΙΙ. ΜΑΘΗΜΑ 4ο
ΦΥΣΙΚΗΣ ΣΤΟΙΧΕΙΩΔΩΝ ΣΩΜΑΤΙΔΙΩΝ ΙΙ ΜΑΘΗΜΑ 4ο Αλληλεπιδράσεις αδρονίου αδρονίου Μελέτη χαρακτηριστικών των ισχυρών αλληλεπιδράσεων (αδρονίων-αδρονίων) Σε θεµελιώδες επίπεδο: αλληλεπιδράσεις µεταξύ quark
Διαβάστε περισσότεραγ 1 6 M = 0.05 F M = 0.05 F M = 0.2 F M = 0.2 F M = 0.05 F M = 0.05 F M = 0.05 F M = 0.2 F M = 0.05 F 2 2 λ τ M = 6000 M = 10000 M = 15000 M = 6000 M = 10000 M = 15000 1 6 τ = 36 1 6 τ = 102 1 6 M = 5000
Διαβάστε περισσότεραOn Numerical Radius of Some Matrices
International Journal of Mathematical Analysis Vol., 08, no., 9-8 HIKARI Ltd, www.m-hikari.com https://doi.org/0.988/ijma.08.75 On Numerical Radius of Some Matrices Shyamasree Ghosh Dastidar Department
Διαβάστε περισσότεραwave energy Superposition of linear plane progressive waves Marine Hydrodynamics Lecture Oblique Plane Waves:
3.0 Marine Hydrodynamics, Fall 004 Lecture 0 Copyriht c 004 MIT - Department of Ocean Enineerin, All rihts reserved. 3.0 - Marine Hydrodynamics Lecture 0 Free-surface waves: wave enery linear superposition,
Διαβάστε περισσότεραOscillating dipole system Suppose we have two small spheres separated by a distance s. The charge on one sphere changes with time and is described by
5 Radiation (Chapte 11) 5.1 Electic dipole adiation Oscillating dipole system Suppose we have two small sphees sepaated by a distance s. The chage on one sphee changes with time and is descibed by q(t)
Διαβάστε περισσότεραExpIntegralE. 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
Διαβάστε περισσότεραInverse 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
Διαβάστε περισσότεραHartree-Fock Theory. Solving electronic structure problem on computers
Hartree-Foc Theory Solving electronic structure problem on computers Hartree product of non-interacting electrons mean field molecular orbitals expectations values one and two electron operators Pauli
Διαβάστε περισσότεραSpace Physics (I) [AP-3044] Lecture 1 by Ling-Hsiao Lyu Oct Lecture 1. Dipole Magnetic Field and Equations of Magnetic Field Lines
Space Physics (I) [AP-344] Lectue by Ling-Hsiao Lyu Oct. 2 Lectue. Dipole Magnetic Field and Equations of Magnetic Field Lines.. Dipole Magnetic Field Since = we can define = A (.) whee A is called the
Διαβάστε περισσότερα100x W=0.1 W=
1 2 abstract: 1 [1] 1970 Furstenberg [2] 1 [3] 1979 Anderson [4] 1 2 1 H = NX n=1 jn >V n < nj); fv n g HjΨ >= EΨ > (ffi n =), ffi n+1 + ffi n 1 + V n ffi n = Effi n,
Διαβάστε περισσότεραMatrices 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
Διαβάστε περισσότεραQuantum Systems: Dynamics and Control 1
Quantum Systems: Dynamics and Control 1 Mazyar Mirrahimi and Pierre Rouchon 3 February 7, 018 1 See the web page: http://cas.ensmp.fr/~rouchon/masterupmc/index.html INRIA Paris, QUANTIC research team 3
Διαβάστε περισσότεραÓ³ 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
Διαβάστε περισσότεραLifting Entry 2. Basic planar dynamics of motion, again Yet another equilibrium glide Hypersonic phugoid motion MARYLAND U N I V E R S I T Y O F
ifting Entry Basic planar dynamics of motion, again Yet another equilibrium glide Hypersonic phugoid motion MARYAN 1 010 avid. Akin - All rights reserved http://spacecraft.ssl.umd.edu ifting Atmospheric
Διαβάστε περισσότεραDark matter from Dark Energy-Baryonic Matter Couplings
Dark matter from Dark Energy-Baryonic Matter Coulings Alejandro Avilés 1,2 1 Instituto de Ciencias Nucleares, UNAM, México 2 Instituto Nacional de Investigaciones Nucleares (ININ) México January 10, 2010
Διαβάστε περισσότεραQuantum Statistical Mechanics (equilibrium) solid state, magnetism black body radiation neutron stars molecules lasers, superuids, superconductors
BYU PHYS 73 Statistical Mechanics Chapter 7: Sethna Professor Manuel Berrondo Quantum Statistical Mechanics (equilibrium) solid state, magnetism black body radiation neutron stars molecules lasers, superuids,
Διαβάστε περισσότερα2. Α ν ά λ υ σ η Π ε ρ ι ο χ ή ς. 3. Α π α ι τ ή σ ε ι ς Ε ρ γ ο δ ό τ η. 4. Τ υ π ο λ ο γ ί α κ τ ι ρ ί ω ν. 5. Π ρ ό τ α σ η. 6.
Π Ε Ρ Ι Ε Χ Ο Μ Ε Ν Α 1. Ε ι σ α γ ω γ ή 2. Α ν ά λ υ σ η Π ε ρ ι ο χ ή ς 3. Α π α ι τ ή σ ε ι ς Ε ρ γ ο δ ό τ η 4. Τ υ π ο λ ο γ ί α κ τ ι ρ ί ω ν 5. Π ρ ό τ α σ η 6. Τ ο γ ρ α φ ε ί ο 1. Ε ι σ α γ ω
Διαβάστε περισσότεραΜοντέρνα Θεωρία Ελέγχου
ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΑΝΟΙΧΤΑ ΑΚΑΔΗΜΑΙΚΑ ΜΑΘΗΜΑΤΑ Ενότητα 5: Διακριτοποίηση συστημάτων Νίκος Καραμπετάκης Άδειες Χρήσης Το παρόν εκπαιδευτικό υλικό υπόκειται σε άδειες χρήσης Creative
Διαβάστε περισσότεραΕισαγωγή στην Ανάλυση και Προσοµοίωση Δυναµικών Συστηµάτων
Εισαγωγή στην Ανάλυση και Προσοµοίωση Δυναµικών Συστηµάτων Control Systems Laboratory Περιγραφή Δυναµικών Συστηµάτων Εξίσωση µεταβολής όγκου Η µεταβολή όγκου ισούται µε τη παροχή υγρού Q που σχετίζεται
Διαβάστε περισσότερα