A thermodynamic characterization of some regularity structures near the subcriticality threshold
|
|
|
- Παρασκευή Ηλιόπουλος
- 5 χρόνια πριν
- Προβολές:
Transcript
1 Nils Berglund SPA 2017 Invited Session: Regularity structures A thermodynamic characterization of some regularity structures near the subcriticality threshold Nils Berglund MAPMO, Université d Orléans, France Moskva / Moscow, July with Christian Kuehn (TU Munich)
2 Fractional Allen Cahn-type SPDEs Thermodynamic characterization of regularity structures July 24, /11 t u = ( ) ρ/2 u + F (u) + ξ
3 Fractional Allen Cahn-type SPDEs Thermodynamic characterization of regularity structures July 24, /11 t u = ( ) ρ/2 u + F (u) + ξ u = u(t, x) R, (t, x) R + T d, d 1 ( ) ρ/2 =: ρ/2 : Fractional Laplacian F polynomial of degree N ξ space-time white noise: E[ξ(t, x)ξ(y, s)] = δ(x y)δ(t s) ξ, ϕ = W ϕ N (0, ϕ 2 L 2 ), E[W ϕ W ϕ ] = ϕ, ϕ
4 Fractional Allen Cahn-type SPDEs Thermodynamic characterization of regularity structures July 24, /11 t u = ( ) ρ/2 u + F (u) + ξ u = u(t, x) R, (t, x) R + T d, d 1 ( ) ρ/2 =: ρ/2 : Fractional Laplacian F polynomial of degree N ξ space-time white noise: E[ξ(t, x)ξ(y, s)] = δ(x y)δ(t s) ξ, ϕ = W ϕ N (0, ϕ 2 L 2 ), E[W ϕ W ϕ ] = ϕ, ϕ Motivations: Better understanding of local subcriticality FitzHugh Nagumo equation [B, Kuehn, EJP 2016] t u = u + u u 3 + v + ξ t v = a 1 u + a 2 v
5 Fractional Allen Cahn-type SPDEs Thermodynamic characterization of regularity structures July 24, /11 t u = ( ) ρ/2 u + F (u) + ξ u = u(t, x) R, (t, x) R + T d, d 1 ( ) ρ/2 =: ρ/2 : Fractional Laplacian F polynomial of degree N ξ space-time white noise: E[ξ(t, x)ξ(y, s)] = δ(x y)δ(t s) ξ, ϕ = W ϕ N (0, ϕ 2 L 2 ), E[W ϕ W ϕ ] = ϕ, ϕ Motivations: Better understanding of local subcriticality FitzHugh Nagumo equation [B, Kuehn, EJP 2016] t u = u + u u 3 + v + ξ t v = δ v + a 1 u + a 2 v
6 Fractional Allen Cahn-type SPDEs Thermodynamic characterization of regularity structures July 24, /11 t u = ( ) ρ/2 u + F (u) + ξ u = u(t, x) R, (t, x) R + T d, d 1 ( ) ρ/2 =: ρ/2 : Fractional Laplacian F polynomial of degree N ξ space-time white noise: E[ξ(t, x)ξ(y, s)] = δ(x y)δ(t s) ξ, ϕ = W ϕ N (0, ϕ 2 L 2 ), E[W ϕ W ϕ ] = ϕ, ϕ Motivations: Better understanding of local subcriticality FitzHugh Nagumo equation [B, Kuehn, EJP 2016] t u = u + u u 3 + v + ξ t v = ρ/2 v + a 1 u + a 2 v
7 Case ρ = 2, N = 3 Thermodynamic characterization of regularity structures July 24, /11 Φ 4 d model: tu = u u 3 + ξ
8 Case ρ = 2, N = 3 Thermodynamic characterization of regularity structures July 24, /11 Φ 4 d model: tu = u u 3 + ξ Mollified noise: ξ ε = ϱ ε ξ with ϱ ε (t, x) = 1 ϱ ( t, x ) ε d+2 ε 2 ε where ϱ compact support, integral 1 Theorem d {2, 3}. choice of renormalisation const C(ε), lim ε 0 C(ε) =, t u ε = u ε + C(ε)u ε (u ε ) 3 + ξ ε admits sequence u ε of local solutions, converging in probability to a limit u as ε 0.
9 Case ρ = 2, N = 3 Thermodynamic characterization of regularity structures July 24, /11 Φ 4 d model: tu = u u 3 + ξ Mollified noise: ξ ε = ϱ ε ξ with ϱ ε (t, x) = 1 ϱ ( t, x ) ε d+2 ε 2 ε where ϱ compact support, integral 1 Theorem d {2, 3}. choice of renormalisation const C(ε), lim ε 0 C(ε) =, t u ε = u ε + C(ε)u ε (u ε ) 3 + ξ ε admits sequence u ε of local solutions, converging in probability to a limit u as ε 0. d = 2: [Da Prato & Debussche 2004] C(ε) = C 1 log(ε 1 ) d = 3: [Hairer 2014], also [Catellier & Chouk], [Kupiainen] C(ε) = C 1 ε 1 + C 2 log(ε 1 ) + C 3
10 Thermodynamic characterization of regularity structures July 24, /11 Mild solutions and Hölder spaces ( t )u = h u = G h ( t )u = F (u) + ξ u = G ξ + G F (u) Which function space?
11 Thermodynamic characterization of regularity structures July 24, /11 Mild solutions and Hölder spaces ( t )u = h u = G h ( t )u = F (u) + ξ u = G ξ + G F (u) Which function space? Hölder spaces C α for f : I R, with I R compact interval: 0 < α < 1: f (x) f (y) C x y α x y α > 1: f C α and f C α 1 f (x) f (y) C x y α α < 0: f distribution, f, ηx δ Cδ α with ηx(y) δ = 1 x y δ η( δ )
12 Mild solutions and Hölder spaces Thermodynamic characterization of regularity structures July 24, /11 ( t )u = h u = G h ( t )u = F (u) + ξ u = G ξ + G F (u) Which function space? Hölder spaces C α for f : I R, with I R compact interval: 0 < α < 1: f (x) f (y) C x y α x y α > 1: f C α and f C α 1 f (x) f (y) C x y α α < 0: f distribution, f, ηx δ Cδ α with ηx(y) δ = 1 x y δ η( δ ) Parabolic scaling C α s : x y t s 1/2 + d i=1 x i y i Facts: 1. α / Z, f Cs α G f Cs α+2 2. ξ Cs α a.s. α < d+2 2 (Schauder)
13 Mild solutions and Hölder spaces Thermodynamic characterization of regularity structures July 24, /11 ( t )u = h u = G h ( t )u = F (u) + ξ u = G ξ + G F (u) Which function space? Hölder spaces C α for f : I R, with I R compact interval: 0 < α < 1: f (x) f (y) C x y α x y α > 1: f C α and f C α 1 f (x) f (y) C x y α α < 0: f distribution, f, ηx δ Cδ α with ηx(y) δ = 1 x y δ η( δ ) Parabolic scaling C α s : x y t s 1/2 + d i=1 x i y i Facts: 1. α / Z, f Cs α G f Cs α+2 (Schauder) 2. ξ Cs α a.s. α < d+2 2 Consequence: G ξ Cs α a.s. α < 2 d 2 0 for d 2
14 Thermodynamic characterization of regularity structures July 24, /11 Regularity structures [Hairer, Inventiones Math. 198, , 2014]: M(u 0, Z ε ) MΨ S M U M R M (u 0, ξ ε ) u ε S M
15 Thermodynamic characterization of regularity structures July 24, /11 Regularity structures [Hairer, Inventiones Math. 198, , 2014]: M(u 0, Z ε ) MΨ S M U M R M (u 0, ξ ε ) S M u ε u = G (ξ ε u 3 ) U = I(Ξ U 3 ) + ϕ1 + polynomial terms U 0 = 0 U 1 = I(Ξ) + ϕ U1 3 = I(Ξ) 3 + 3ϕI(Ξ) 2 + 3ϕ 2 I(Ξ) + ϕ U 2 = I(Ξ) I(I(Ξ) 3 ) 3ϕI(I(Ξ) 2 ) 3ϕ 2 I(I(Ξ)) + ϕ1 +...
16 Thermodynamic characterization of regularity structures July 24, /11 Regularity structures [Hairer, Inventiones Math. 198, , 2014]: M(u 0, Z ε ) MΨ S M U M R M (u 0, ξ ε ) S M u ε u = G (ξ ε u 3 ) U = I(Ξ U 3 ) + ϕ1 + polynomial terms U 0 = 0 U 1 = I(Ξ) + ϕ U1 3 = I(Ξ) 3 + 3ϕI(Ξ) 2 + 3ϕ 2 I(Ξ) + ϕ U 2 = I(Ξ) I(I(Ξ) 3 ) 3ϕI(I(Ξ) 2 ) 3ϕ 2 I(I(Ξ)) + ϕ1 +...
17 Thermodynamic characterization of regularity structures July 24, /11 Regularity structures [Hairer, Inventiones Math. 198, , 2014]: M(u 0, Z ε ) MΨ S M U M R M (u 0, ξ ε ) S M u ε u = G (ξ ε u 3 ) U = I(Ξ U 3 ) + ϕ1 + polynomial terms U 0 = 0 U 1 = I(Ξ) + ϕ U1 3 = I(Ξ) 3 + 3ϕI(Ξ) 2 + 3ϕ 2 I(Ξ) + ϕ U 2 = I(Ξ) I(I(Ξ) 3 ) 3ϕI(I(Ξ) 2 ) 3ϕ 2 I(I(Ξ)) + ϕ =: 3ϕ 3ϕ 2 + ϕ1 +...
18 Regularity structures [Hairer, Inventiones Math. 198, , 2014]: M(u 0, Z ε ) MΨ S M U M R M (u 0, ξ ε ) S M u ε u = G (ξ ε u 3 ) U = I(Ξ U 3 ) + ϕ1 + polynomial terms U 0 = 0 U 1 = I(Ξ) + ϕ U1 3 = I(Ξ) 3 + 3ϕI(Ξ) 2 + 3ϕ 2 I(Ξ) + ϕ U 2 = I(Ξ) I(I(Ξ) 3 ) 3ϕI(I(Ξ) 2 ) 3ϕ 2 I(I(Ξ)) + ϕ =: 3ϕ 3ϕ 2 + ϕ d 2 5 3d 2 4 d 3 d 2 0 Locally subcritical: Hölder exponents are bdd below Thermodynamic characterization of regularity structures July 24, /11
19 Thermodynamic characterization of regularity structures July 24, /11 Case of fractional Laplacian 0 < ρ < 2, ρ/2 := ( ) ρ/2 generator of a Lévy process with kernel G ρ (t, x) = 1 (2π) d e i x ξ e t ξ ρ dξ R d
20 Thermodynamic characterization of regularity structures July 24, /11 Case of fractional Laplacian 0 < ρ < 2, ρ/2 := ( ) ρ/2 generator of a Lévy process with kernel G ρ (t, x) = 1 (2π) d e i x ξ e t ξ ρ dξ R d Easy to check: 1. s = (ρ, 1,..., 1) G ρ regularising of order ρ: f C α s, α + ρ / Z G ρ f C α+ρ s
21 Thermodynamic characterization of regularity structures July 24, /11 Case of fractional Laplacian 0 < ρ < 2, ρ/2 := ( ) ρ/2 generator of a Lévy process with kernel G ρ (t, x) = 1 (2π) d e i x ξ e t ξ ρ dξ R d Easy to check: 1. s = (ρ, 1,..., 1) G ρ regularising of order ρ: f C α s, α + ρ / Z G ρ f C α+ρ s 2. ξ C α s a.s. α < ρ+d 2
22 Thermodynamic characterization of regularity structures July 24, /11 Case of fractional Laplacian 0 < ρ < 2, ρ/2 := ( ) ρ/2 generator of a Lévy process with kernel G ρ (t, x) = 1 (2π) d e i x ξ e t ξ ρ dξ R d Easy to check: 1. s = (ρ, 1,..., 1) G ρ regularising of order ρ: f C α s, α + ρ / Z G ρ f C α+ρ s 2. ξ C α s a.s. α < ρ+d 2 3. t u = ρ/2 u + F (u) degree N + ξ loc. subcritical ρ > ρ c = d N 1 N+1
23 Thermodynamic characterization of regularity structures July 24, /11 Case of fractional Laplacian 0 < ρ < 2, ρ/2 := ( ) ρ/2 generator of a Lévy process with kernel G ρ (t, x) = 1 (2π) d e i x ξ e t ξ ρ dξ R d Easy to check: 1. s = (ρ, 1,..., 1) G ρ regularising of order ρ: f C α s, α + ρ / Z G ρ f C α+ρ s 2. ξ C α s a.s. α < ρ+d 2 3. t u = ρ/2 u + F (u) degree N + ξ loc. subcritical ρ > ρ c = d N 1 N+1 Idea: Fixed point equ U = I(Ξ + F (U)) + ϕ Idea: Ξ s = ρ+d 2 κ =: α 0 Idea: I(Ξ) N s = N(α 0 + ρ) = N 2 (ρ d) Nκ Idea: I(Ξ) N s > Ξ s ρ > ρ c Idea: then induction on fixed point application
24 Thermodynamic characterization of regularity structures July 24, /11 Model space U = I(Ξ + F (U)) + ϕ U F : symbols representing solution U F F U F : symbols representing equation (i.e. Ξ + F (U))
25 Thermodynamic characterization of regularity structures July 24, /11 Model space U = I(Ξ + F (U)) + ϕ U F : symbols representing solution U F F U F : symbols representing equation (i.e. Ξ + F (U)) given by induction W 0 = U 0 = and W m = W m 1 U m 1 Um 1 N {Ξ} U m = I(W m ) {X k } with AB := {ττ : τ A, τ B} Then U F = U m, F F = m U m ) m 0 m 0(W
26 Thermodynamic characterization of regularity structures July 24, /11 Model space U = I(Ξ + F (U)) + ϕ U F : symbols representing solution U F F U F : symbols representing equation (i.e. Ξ + F (U)) given by induction W 0 = U 0 = and W m = W m 1 U m 1 Um 1 N {Ξ} U m = I(W m ) {X k } with AB := {ττ : τ A, τ B} Then U F = U m, F F = m U m ) m 0 m 0(W Questions: Let A F = { τ s : τ F F } 1. Estimate h F = #(A F R ) (number of negative Hölder exponents) 2. Estimate c F = #{τ F F : τ s < 0} (number of singular symbols)
27 Thermodynamic characterization of regularity structures July 24, /11 Number of negative Hölder exponents h F = #(A F R ) Theorem 1 ρ + d 1 h F 1 + N + 1 ρ ρ c (ρ + d)dn N ρ ρ c
28 Thermodynamic characterization of regularity structures July 24, /11 Number of negative Hölder exponents h F = #(A F R ) Theorem 1 ρ + d 1 h F 1 + N + 1 ρ ρ c Proof: τ s = ρ+d 2 p(τ) + q(τ)ρ + k s(τ) O(κ) p = #Ξ, q = #I, 0 k s = polynomial exp. D 0 (U) = {(p(τ), q(τ)): τ U} N 2 0 (ρ + d)dn N ρ ρ c
29 Thermodynamic characterization of regularity structures July 24, /11 Number of negative Hölder exponents h F = #(A F R ) Theorem 1 ρ + d 1 h F 1 + N + 1 ρ ρ c (ρ + d)dn N ρ ρ c D 0 (W 3 ) for N = d = 3 ρ = 2 I ρ = Ξ ρ = 9 5 ρ = ρ c = 3 2
30 Thermodynamic characterization of regularity structures July 24, /11 Number of negative Hölder exponents h F = #(A F R ) Theorem 1 ρ + d 1 h F 1 + N + 1 ρ ρ c Proof: τ s = ρ+d 2 p(τ) + q(τ)ρ + k s(τ) O(κ) p = #Ξ, q = #I, 0 k s = polynomial exp. (ρ + d)dn N ρ ρ c D 0 (U) = {(p(τ), q(τ)): τ U} N 2 0 (p, q ) D 0 (U n ) = convex env. of nd 0 (U) N 2 0 lim m D 0 (U m ) = truncated cone τ s < 0 p = 1 + N 1 N q & τ triangle h F = number of lattice points in triangle q = (ρ+d)n (N+1)(ρ ρ + O(κ) c) q p q = ρ+d 2 p + qρ = 0 N N 1 (p 1)
31 Thermodynamic characterization of regularity structures July 24, /11 Number of symbols c F = #{τ F F : τ s < 0} Theorem 2 C N (ρ ρ c) 3/2 e β Nd/(ρ ρ c) c F C + N (ρ ρ c) 3/2 e β Nd/(ρ ρ c)
32 Thermodynamic characterization of regularity structures July 24, /11 Number of symbols c F = #{τ F F : τ s < 0} Theorem 2 C N (ρ ρ c) 3/2 e β Nd/(ρ ρ c) c F C + N (ρ ρ c) 3/2 e β Nd/(ρ ρ c) Case N = 2: τ trees of degree 3, p leaves, q edges d i := # vertices of degree i d 1 + d 2 + d 3 = q + 1 d 1 + 2d 2 + 3d 3 = 2q d 1 = p + 1 {deg =1} I Ξ
33 Thermodynamic characterization of regularity structures July 24, /11 Number of symbols c F = #{τ F F : τ s < 0} Theorem 2 C N (ρ ρ c) 3/2 e β Nd/(ρ ρ c) c F C + N (ρ ρ c) 3/2 e β Nd/(ρ ρ c) Case N = 2: τ trees of degree 3, p leaves, q edges d i := # vertices of degree i d 1 + d 2 + d 3 = q + 1 d 1 + 2d 2 + 3d 3 = 2q d 1 = p + 1 {deg =1} q = 2n binary tree with q + 1 vertices q = 2n + 1 binary tree with q + 2 vertices minus one edge I Ξ
34 Number of symbols c F = #{τ F F : τ s < 0} Theorem 2 C N (ρ ρ c) 3/2 e β Nd/(ρ ρ c) c F C + N (ρ ρ c) 3/2 e β Nd/(ρ ρ c) Case N = 2: τ trees of degree 3, p leaves, q edges d i := # vertices of degree i d 1 + d 2 + d 3 = q + 1 d 1 + 2d 2 + 3d 3 = 2q d 1 = p + 1 {deg =1} q = 2n binary tree with q + 1 vertices q = 2n + 1 binary tree with q + 2 vertices minus one edge One has to count trees up to homeomorphism (1/ )n Wedderburn Etherington numbers W n c [Otter 1948] n 3/2 Thermodynamic characterization of regularity structures July 24, /11 I Ξ
35 Thermodynamic characterization of regularity structures July 24, /11 Statistical properties Ω = {τ F F : τ s < 0}, P uniform measure Properties of random variables X : Ω R when ρ ρ c?
36 Thermodynamic characterization of regularity structures July 24, /11 Statistical properties Ω = {τ F F : τ s < 0}, P uniform measure Properties of random variables X : Ω R when ρ ρ c? Case X = Q = #I: E[Q/q ] = 1 + O(ρ ρ c ) et Var[Q/q ] = O((ρ ρ c ) 2 ) lim ρ ρ c (ρ ρ c ) log P{Q/q x} = β N d(1 x) x [0, 1] P{Q / NN} e γ/(ρ ρc)
![Case X = Q = #I: E[Q/q ] = 1 + O(ρ ρ c ) et Var[Q/q ] = O((ρ ρ c ) 2 ) lim ρ ρ c (ρ ρ c ) log P{Q/q x} = β N d(1 x) x [0, 1] P{Q / NN} e γ/(ρ ρc)](/docs-images/93/113919993/images/37-1.jpg "Other interesting random variables: Number P of Ξ: function of Q Hölder exponent: concentrated in 0 Degree distribution: close to ( N 1 N, 0,.")
37 Thermodynamic characterization of regularity structures July 24, /11 Statistical properties Ω = {τ F F : τ s < 0}, P uniform measure Properties of random variables X : Ω R when ρ ρ c? Case X = Q = #I: E[Q/q ] = 1 + O(ρ ρ c ) et Var[Q/q ] = O((ρ ρ c ) 2 ) lim ρ ρ c (ρ ρ c ) log P{Q/q x} = β N d(1 x) x [0, 1] P{Q / NN} e γ/(ρ ρc) Other interesting random variables: Number P of Ξ: function of Q Hölder exponent: concentrated in 0 Degree distribution: close to ( N 1 N, 0,..., 0, 1 N ) Height and diameter [Broutin & Flajolet]: of order 1/ ρ ρ c N = d = 3, ρ {1.8, 1.75, 1.7, 1.76, 1.6, 1.59}
38 Thermodynamic characterization of regularity structures July 24, /11 Renormalisation t u ε = ρ/2 u ε + C(ε)u ε (u ε ) 3 + ξ ε
39 Thermodynamic characterization of regularity structures July 24, /11 Renormalisation t u ε = ρ/2 u ε + C(ε)u ε (u ε ) 3 + ξ ε BPHZ theory [Bruned, Hairer & Zambotti 2016, Chandra & Hairer 2016]: We expect C(ε) ε τ s (ε 0 = log(ε 1 )) τ F F : τ s<0
40 Thermodynamic characterization of regularity structures July 24, /11 Renormalisation t u ε = ρ/2 u ε + C(ε)u ε (u ε ) 3 + ξ ε BPHZ theory [Bruned, Hairer & Zambotti 2016, Chandra & Hairer 2016]: We expect C(ε) ε τ s (ε 0 = log(ε 1 )) τ F F : τ s<0 = c F E[ε τ s ]
41 Thermodynamic characterization of regularity structures July 24, /11 Renormalisation t u ε = ρ/2 u ε + C(ε)u ε (u ε ) 3 + ξ ε BPHZ theory [Bruned, Hairer & Zambotti 2016, Chandra & Hairer 2016]: We expect C(ε) ε τ s (ε 0 = log(ε 1 )) τ F F : τ s<0 = c F E[ε τ s ] P{ τ s = h} e γ Nh/(ρ ρ c) γ N = N+1 N β N ρ+d 2 < h < 0 number of elements number of elements rho = homogeneity rho = homogeneity number of elements number of elements rho = homogeneity rho = homogeneity number of elements number of elements rho = homogeneity rho = homogeneity
42 Thermodynamic characterization of regularity structures July 24, /11 Renormalisation t u ε = ρ/2 u ε + C(ε)u ε (u ε ) 3 + ξ ε BPHZ theory [Bruned, Hairer & Zambotti 2016, Chandra & Hairer 2016]: We expect C(ε) ε τ s (ε 0 = log(ε 1 )) τ F F : τ s<0 = c F E[ε τ s ] P{ τ s = h} e γ Nh/(ρ ρ c) γ N = N+1 N β N ρ+d 2 < h < 0 number of elements number of elements rho = homogeneity rho = 1.65 number of elements number of elements rho = homogeneity rho = homogeneity homogeneity c F log(ε 1 ) if ε > e γ N/(ρ ρ c) ( ) C(ε) ε (d ρ) c F if ε < e γ N/(ρ ρ c) e γ N/(ρ ρ c) number of elements number of elements rho = homogeneity rho = homogeneity
43 Thermodynamic characterization of regularity structures July 24, /11 References M. Hairer, A theory of regularity structures, Inv. Math. 198, (2014) M. Hairer, Introduction to Regularity Structures, lecture notes (2013) A. Chandra, H. Weber, Stochastic PDEs, regularity structures, and interacting particle systems, Annales Mathématiques de la Faculté des Sciences de Toulouse (in press), arxiv/ N. B., C. Kuehn, Regularity structures and renormalisation of FitzHugh Nagumo SPDEs in three space dimensions, Elec J Prob 21 (18):1 48 (2016) N. B., C. Kuehn, Model spaces of regularity structures in space-fractional SPDEs, J Statist Phys (168): (2017) Y. Bruned, M. Hairer, L. Zambotti, Algebraic renormalisation of regularity structures, arxiv/ A. Chandra, M. Hairer, An analytic BPHZ theorem for regularity structures, arxiv/ M. Hairer, An analyst s take on the BPHZ theorem, arxiv/
44 Advertisement At CIRM, Marseille Luminy, France: Thermodynamic characterization of regularity structures July 24, /11
Structures de régularité et mécanique statistique
Nils Berglund nils.berglund@univ-orleans.fr http://www.univ-orleans.fr/mapmo/membres/berglund/ Journées de Probabilités Structures de régularité et mécanique statistique Nils Berglund MAPMO, Université
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
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
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,
The Pohozaev identity for the fractional Laplacian
The Pohozaev identity for the fractional Laplacian Xavier Ros-Oton Departament Matemàtica Aplicada I, Universitat Politècnica de Catalunya (joint work with Joaquim Serra) Xavier Ros-Oton (UPC) The Pohozaev
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
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
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)
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
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
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
On a four-dimensional hyperbolic manifold with finite volume
BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Numbers 2(72) 3(73), 2013, Pages 80 89 ISSN 1024 7696 On a four-dimensional hyperbolic manifold with finite volume I.S.Gutsul Abstract. In
2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits.
EAMCET-. THEORY OF EQUATIONS PREVIOUS EAMCET Bits. Each of the roots of the equation x 6x + 6x 5= are increased by k so that the new transformed equation does not contain term. Then k =... - 4. - Sol.
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
Continuous Distribution Arising from the Three Gap Theorem
Continuous Distribution Arising from the Three Gap Theorem Geremías Polanco Encarnación Elementary Analytic and Algorithmic Number Theory Athens, Georgia June 8, 2015 Hampshire college, MA 1 / 24 Happy
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
Local Approximation with Kernels
Local Approximation with Kernels Thomas Hangelbroek University of Hawaii at Manoa 5th International Conference Approximation Theory, 26 work supported by: NSF DMS-43726 A cubic spline example Consider
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
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
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) =
HW 3 Solutions 1. a) I use the auto.arima R function to search over models using AIC and decide on an ARMA(3,1)
HW 3 Solutions a) I use the autoarima R function to search over models using AIC and decide on an ARMA3,) b) I compare the ARMA3,) to ARMA,0) ARMA3,) does better in all three criteria c) The plot of the
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
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
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
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
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
Global 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)
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 :
1. (a) (5 points) Find the unit tangent and unit normal vectors T and N to the curve. r(t) = 3cost, 4t, 3sint
1. a) 5 points) Find the unit tangent and unit normal vectors T and N to the curve at the point P, π, rt) cost, t, sint ). b) 5 points) Find curvature of the curve at the point P. Solution: a) r t) sint,,
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
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
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
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
The 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,
Mellin transforms and asymptotics: Harmonic sums
Mellin tranform and aymptotic: Harmonic um Phillipe Flajolet, Xavier Gourdon, Philippe Duma Die Theorie der reziproen Funtionen und Integrale it ein centrale Gebiet, welche manche anderen Gebiete der Analyi
Homomorphism 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
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
F19MC2 Solutions 9 Complex Analysis
F9MC Solutions 9 Complex Analysis. (i) Let f(z) = eaz +z. Then f is ifferentiable except at z = ±i an so by Cauchy s Resiue Theorem e az z = πi[res(f,i)+res(f, i)]. +z C(,) Since + has zeros of orer at
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
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
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)
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
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
The 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,
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
SOME PROPERTIES OF FUZZY REAL NUMBERS
Sahand Communications in Mathematical Analysis (SCMA) Vol. 3 No. 1 (2016), 21-27 http://scma.maragheh.ac.ir SOME PROPERTIES OF FUZZY REAL NUMBERS BAYAZ DARABY 1 AND JAVAD JAFARI 2 Abstract. In the mathematical
On the k-bessel Functions
International Mathematical Forum, Vol. 7, 01, no. 38, 1851-1857 On the k-bessel Functions Ruben Alejandro Cerutti Faculty of Exact Sciences National University of Nordeste. Avda. Libertad 5540 (3400) Corrientes,
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
forms This gives Remark 1. How to remember the above formulas: Substituting these into the equation we obtain with
Week 03: C lassification of S econd- Order L inear Equations In last week s lectures we have illustrated how to obtain the general solutions of first order PDEs using the method of characteristics. We
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
Aquinas College. Edexcel Mathematical formulae and statistics tables DO NOT WRITE ON THIS BOOKLET
Aquinas College Edexcel Mathematical formulae and statistics tables DO NOT WRITE ON THIS BOOKLET Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical
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
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
Figure A.2: MPC and MPCP Age Profiles (estimating ρ, ρ = 2, φ = 0.03)..
Supplemental Material (not for publication) Persistent vs. Permanent Income Shocks in the Buffer-Stock Model Jeppe Druedahl Thomas H. Jørgensen May, A Additional Figures and Tables Figure A.: Wealth and
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:
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.
John Nash. Παύλος Στ. Εφραιµίδης. Τοµέας Λογισµικού και Ανάπτυξης Εφαρµογών Τµήµα Ηλεκτρολόγων Μηχανικών και Μηχανικών Υπολογιστών
Παύλος Στ. Εφραιµίδης Τοµέας Λογισµικού και Ανάπτυξης Εφαρµογών Τµήµα Ηλεκτρολόγων Μηχανικών και Μηχανικών Υπολογιστών ορισµένα αποτελέσµατα του τα σηµεία ισορροπίας Nash (NE Nash Equilibrium) ύπαρξη σηµείου
Affine Weyl Groups. Gabriele Nebe. Summerschool GRK 1632, September Lehrstuhl D für Mathematik
Affine Weyl Groups Gabriele Nebe Lehrstuhl D für Mathematik Summerschool GRK 1632, September 2015 Crystallographic root systems. Definition A crystallographic root system Φ is a finite set of non zero
Spherical 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
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
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 +
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
Markov 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 /
D 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
ON NEGATIVE MOMENTS OF CERTAIN DISCRETE DISTRIBUTIONS
Pa J Statist 2009 Vol 25(2), 135-140 ON NEGTIVE MOMENTS OF CERTIN DISCRETE DISTRIBUTIONS Masood nwar 1 and Munir hmad 2 1 Department of Maematics, COMSTS Institute of Information Technology, Islamabad,
Mock 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
Approximation of dynamic boundary condition: The Allen Cahn equation
1 Approximation of dynamic boundary condition: The Allen Cahn equation Matthias Liero I.M.A.T.I. Pavia and Humboldt-Universität zu Berlin 6 th Singular Days 2010 Berlin Introduction Interfacial dynamics
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
EXISTENCE OF POSITIVE SOLUTIONS FOR SINGULAR FRACTIONAL DIFFERENTIAL EQUATIONS
Electronic Journal of Differential Equations, Vol. 28(28), No. 146, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) EXISTENCE
SCITECH 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
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
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
The circle theorem and related theorems for Gauss-type quadrature rules
OP.circle p. / The circle theorem and related theorems for Gauss-type quadrature rules Walter Gautschi wxg@cs.purdue.edu Purdue University OP.circle p. 2/ Web Site http : //www.cs.purdue.edu/ archives/22/wxg/codes
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
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
Parametrized Surfaces
Parametrized Surfaces Recall from our unit on vector-valued functions at the beginning of the semester that an R 3 -valued function c(t) in one parameter is a mapping of the form c : I R 3 where I is some
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
Queensland University of Technology Transport Data Analysis and Modeling Methodologies
Queensland University of Technology Transport Data Analysis and Modeling Methodologies Lab Session #7 Example 5.2 (with 3SLS Extensions) Seemingly Unrelated Regression Estimation and 3SLS A survey of 206
Lecture 7: Overdispersion in Poisson regression
Lecture 7: Overdispersion in Poisson regression Claudia Czado TU München c (Claudia Czado, TU Munich) ZFS/IMS Göttingen 2004 0 Overview Introduction Modeling overdispersion through mixing Score test for
26 28 Find an equation of the tangent line to the curve at the given point Discuss the curve under the guidelines of Section
SECTION 5. THE NATURAL LOGARITHMIC FUNCTION 5. THE NATURAL LOGARITHMIC FUNCTION A Click here for answers. S Click here for solutions. 4 Use the Laws of Logarithms to epand the quantit.. ln ab. ln c. ln
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
( 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
1. 3. ([12], Matsumura[13], Kikuchi[10] ) [12], [13], [10] ( [12], [13], [10]
![1. 3. ([12], Matsumura[13], Kikuchi[10] ) [12], [13], [10] ( [12], [13], [10]](/thumbs/103/159380891.jpg "1. 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]
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
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
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
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 >
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,
Iterated trilinear fourier integrals with arbitrary symbols
Cornell University ICM 04, Satellite Conference in Harmonic Analysis, Chosun University, Gwangju, Korea August 6, 04 Motivation the Coifman-Meyer theorem with classical paraproduct(979) B(f, f )(x) :=
CHAPTER 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.
Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1
Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the
SOLVING CUBICS AND QUARTICS BY RADICALS
SOLVING CUBICS AND QUARTICS BY RADICALS The purpose of this handout is to record the classical formulas expressing the roots of degree three and degree four polynomials in terms of radicals. We begin with
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
Web-based supplementary materials for Bayesian Quantile Regression for Ordinal Longitudinal Data
Web-based supplementary materials for Bayesian Quantile Regression for Ordinal Longitudinal Data Rahim Alhamzawi, Haithem Taha Mohammad Ali Department of Statistics, College of Administration and Economics,
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
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
ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Αν κάπου κάνετε κάποιες υποθέσεις να αναφερθούν στη σχετική ερώτηση. Όλα τα αρχεία που αναφέρονται στα προβλήματα βρίσκονται στον ίδιο φάκελο με το εκτελέσιμο
MA 342N Assignment 1 Due 24 February 2016
M 342N ssignment Due 24 February 206 Id: 342N-s206-.m4,v. 206/02/5 2:25:36 john Exp john. Suppose that q, in addition to satisfying the assumptions from lecture, is an even function. Prove that η(λ = 0,
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.
Integrals in cylindrical, spherical coordinates (Sect. 15.7)
Integrals in clindrical, spherical coordinates (Sect. 5.7 Integration in spherical coordinates. Review: Clindrical coordinates. Spherical coordinates in space. Triple integral in spherical coordinates.