# Orthogonal systems and semigroups

Μέγεθος: px
Εμφάνιση ξεκινά από τη σελίδα:

## Transcript

1 Orthogonal systems and semigroups José Luis Torrea Orthonet Logroño, 23 Febrero 213 () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

2 Jorge Betancor, Raquel Crescimbeni, Eleonor Harboure Teresa Martinez, Ricardo Testoni, Peter Sjögren Orthogonal systems and semigroups Juan Carlos Fariña, José Luis Torrea Oscar Ciaurri Luz Roncal, Juan Luis Varona, Juan Carlos Fariña Beatriz Viviani, Silvia Harzstein, Orthonet Lourdes Rodriguez-Mesa F. J. Martín-Reyes, Logroño, 23 Febrero 213 A. De La Torre, Gustavo Garrigós Alejandro Sanabria, Zhang Chao, Ma Tao Pablo Stinga, Beatriz Viviani, Roberto Macías Mario Pérez, Bruno Bongioanni, Sundaram Thangavelu Carlos Segovia, Cristian Gutierrez, () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

3 Fourier sums Poisson sums f (θ) = k a k e ikθ () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

4 Fourier sums Poisson sums f (θ) = k a k e ikθ P r f (θ) = k r k a k e ikθ = 1 + k> r k a k e ikθ + k> r k a k e ikθ }{{} = 1 + a k z k + r k a k z k = U(z) (z=re iθ ) k> k> () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

5 Fourier sums Poisson sums f (θ) = k a k e ikθ P r f (θ) = k r k a k e ikθ = 1 + k> r k a k e ikθ + k> r k a k e ikθ }{{} = 1 + a k z k + r k a k z k = U(z) (z=re iθ ) k> k> Harmonic. ( r 2 2 r + r r + 2 θ) Pr f (θ) = z z U = () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

6 Fourier sums Poisson sums f (θ) = k a k e ikθ P r f (θ) = k r k a k e ikθ = 1 + k> r k a k e ikθ + k> r k a k e ikθ }{{} = 1 + a k z k + r k a k z k = U(z) (z=re iθ ) k> k> Harmonic. ( r 2 2 r + r r + 2 θ) Pr f (θ) = z z U = (r r ) = r r with respect to dµ(r) = dr r en [, 1]. ( θ ) = θ with respect to a dθ. Decomposition: [ ] r 2 r 2 + r r + θ 2 = (r r ) (r r ) + ( θ ) ( θ ) () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

7 Conjugate harmonic function (Fourier series) f (θ) = k a k e ikθ () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

8 Conjugate harmonic function (Fourier series) f (θ) = k a k e ikθ Q r f (θ) = i k = i k> sign k r k a k e ikθ r k a k e ikθ + i k> r k a k e ikθ }{{} = i a k z k + i r k a k z k = V (z) (z=re iθ ) k> k> () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

9 Conjugate harmonic function (Fourier series) f (θ) = k a k e ikθ Q r f (θ) = i k = i k> sign k r k a k e ikθ r k a k e ikθ + i k> r k a k e ikθ }{{} = i a k z k + i r k a k z k = V (z) (z=re iθ ) k> k> Harmonic. ( r 2 2 r + r r + 2 θ) Qr f (θ) =. z z V =. () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

10 Cauchy Riemann equations ( Fourier series) P r f (θ) = k r k a k e ikθ = 1 + k> a k z k + k> r k a k z k = U(z) Q r f (θ) = i sign k r k a k e ikθ = i a k z k + i r k a k z k = V (z) k k> k> () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

11 Cauchy Riemann equations ( Fourier series) P r f (θ) = k r k a k e ikθ = 1 + k> a k z k + k> r k a k z k = U(z) Q r f (θ) = i sign k r k a k e ikθ = i a k z k + i r k a k z k = V (z) k k> k> F (z) = U(z) + iv (z) = k> a k z k is holomorfic: z F = () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

12 Cauchy Riemann equations ( Fourier series) P r f (θ) = k r k a k e ikθ = 1 + k> a k z k + k> r k a k z k = U(z) Q r f (θ) = i sign k r k a k e ikθ = i a k z k + i r k a k z k = V (z) k k> k> F (z) = U(z) + iv (z) = k> a k z k is holomorfic: z F = Cauchy-Riemann (Fourier series) θ (P r f )(θ) = r r (Q r f )(θ) ( ) r r (P r f )(θ) = θ (Q r f )(θ) i.e. (r r ) (P r f )(θ) = ( θ ) (Q r f )(θ) () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

13 On R f (x) = f (ξ)e ixξ dξ R () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

14 On R f (x) = R f (ξ)e ixξ dξ P t f (x) = R = }{{} (z=t ix) e t ξ f (ξ) e iξ x dξ e zξ f (ξ)dξ + e zξ f ( ξ)dξ = U(z) () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

15 On R f (x) = R f (ξ)e ixξ dξ P t f (x) = R = }{{} (z=t ix) e t ξ f (ξ) e iξ x dξ e zξ f (ξ)dξ + e zξ f ( ξ)dξ = U(z) Harmonic. ( 2 t + 2 x ) Pt f (x) = z z U = () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

16 On R f (x) = R f (ξ)e ixξ dξ P t f (x) = R = }{{} (z=t ix) e t ξ f (ξ) e iξ x dξ e zξ f (ξ)dξ + e zξ f ( ξ)dξ = U(z) Harmonic. ( 2 t + 2 x ) Pt f (x) = z z U = ( x ) = x with respect to dx ( t ) = t with respect to dt Decomposition. [ ] t 2 + x 2 = ( t ) ( t ) + ( x ) ( x ) () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

17 Conjugate harmonic function (line) f (x) = f (ξ)e ixξ dξ R () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

18 Conjugate harmonic function (line) f (x) = R f (ξ)e ixξ dξ Q t f (x) = i R }{{} = i (z=t ix) sign ξ e t ξ f (ξ) e iξ x dξ e zξ f (ξ)dξ + i e zξ f ( ξ)dξ = V (z) () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

19 Conjugate harmonic function (line) f (x) = R f (ξ)e ixξ dξ Q t f (x) = i R }{{} = i (z=t ix) sign ξ e t ξ f (ξ) e iξ x dξ e zξ f (ξ)dξ + i e zξ f ( ξ)dξ = V (z) Harmonic. ( 2 t + 2 x ) Qt f (x) = z z V = () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

20 Cauchy Riemann equations (line) P t f (x) = e t ξ f (ξ) e iξ x dξ = R Q t f (x) = i sign ξ e t ξ f (ξ) e iξ x dξ = i R e zξ f (ξ)dξ + e zξ f ( ξ)dξ = U(z) e zξ f (ξ)dξ + i e zξ f ( ξ)dξ = V (z) () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

21 Cauchy Riemann equations (line) P t f (x) = e t ξ f (ξ) e iξ x dξ = R Q t f (x) = i sign ξ e t ξ f (ξ) e iξ x dξ = i R e zξ f (ξ)dξ + e zξ f ( ξ)dξ = U(z) e zξ f (ξ)dξ + i e zξ f ( ξ)dξ = V (z) F (z) = U(z) + iv (z) is holomorphic: z F = () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

22 Cauchy Riemann equations (line) P t f (x) = e t ξ f (ξ) e iξ x dξ = R Q t f (x) = i sign ξ e t ξ f (ξ) e iξ x dξ = i R e zξ f (ξ)dξ + e zξ f ( ξ)dξ = U(z) e zξ f (ξ)dξ + i e zξ f ( ξ)dξ = V (z) F (z) = U(z) + iv (z) is holomorphic: z F = Cauchy Riemann (line) x P t f (x) = t Q t f (x) ( ) t P t f (x) = x Q t f (x) i.e. ( t ) P t f (x) = ( x ) Q t f (x) () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

23 Orthogonal polynomials (Hermite) Hermite Polynomials on R: H n (x) = ( 1) n e x 2 /2 d n 2 dx n e x /2. () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

24 Orthogonal polynomials (Hermite) Hermite Polynomials on R: H n (x) = ( 1) n e x 2 /2 d n 2 dx n e x /2. f (x) = n a n H n (x) () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

25 Orthogonal polynomials (Hermite) Hermite Polynomials on R: H n (x) = ( 1) n e x 2 /2 d n 2 dx n e x /2. f (x) = n a n H n (x) Poisson sums : P r f (x) = r n a n H n (x)... (r = e t ) n= () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

26 Orthogonal polynomials (Hermite) Hermite Polynomials on R: H n (x) = ( 1) n e x 2 /2 d n 2 dx n e x /2. f (x) = n a n H n (x) Poisson sums : P r f (x) = r n a n H n (x)... (r = e t ) n= There is a kernel : [ P t f (x) = e tn a n H n (x) = e tn n= = R n= [ ] e tn H n (y) H n (x) f (y)e y 2 dy = n= R ] f (y)h n (y)e y 2 dy H n (x) R K t (x, y)f (y)e y 2 dy... () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

27 Literature B. Muckenhoupt, Poisson integrals for Hermite and Laguerre expansions, Trans. Amer. Math. Soc. 118 (1965), B. Muckenhoupt, Hermite conjugate expansions, Trans. Amer. Math. Soc. 139 (1969), B. Muckenhoupt and E. M. Stein Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc. 118 (1965), () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

28 Literature () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

29 Understanding Muckenhoupt Two alternatives following Muckenhoupt P t f (x) = n e t2n a n H n (x) versus P t f (x) = n e t 2n a n H n (x). () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

30 Understanding Muckenhoupt Two alternatives following Muckenhoupt P t f (x) = n e t2n a n H n (x) versus P t f (x) = n e t 2n a n H n (x). The second satisfies ( 2 t + x 2 ) 2x x (Pt f )(x) = ( t 2 ) + ( x 2x) x (Pt f )(x) ] = [( t ) ( t ) + ( x ) x (P t f )(x) = () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

31 Understanding Muckenhoupt Two alternatives following Muckenhoupt P t f (x) = n e t2n a n H n (x) versus P t f (x) = n e t 2n a n H n (x). The second satisfies ( 2 t + x 2 ) 2x x (Pt f )(x) = ( t 2 ) + ( x 2x) x (Pt f )(x) ] = [( t ) ( t ) + ( x ) x (P t f )(x) = ( x ) = ( x 2x) Adjoint with respect to measure dγ(x) = e x 2 dx () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

32 Q t f (x) = n Understanding Muckenhoupt Moreover 2n e t 2n a n H n 1 (x) satisfies ( 2 t + x ( x 2x) ) (Q t f )(x) = [ ( t )( t ) + x ( x ) ] (Q t f )(x) = () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

33 Understanding Muckenhoupt Moreover 2n e t 2n a n H n 1 (x) Q t f (x) = n satisfies ( 2 t + x ( x 2x) ) (Q t f )(x) = [ ( t )( t ) + x ( x ) ] (Q t f )(x) = Cauchy Riemann equations (Hermite) x P t f (x) = t Q t f (x) ( ) t P t f (x) = ( x 2x)Q t f (x), i.e. ( t ) P t f (x) = ( x ) Q t f (x) () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

34 Common path? In all the cases, lim Q tf (x) = t () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

35 Common path? In all the cases, Hence Q t f (x) = t s Q s f (x) ds = lim Q tf (x) = t t x P s f (x) ds = x P s f (x) ds t () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

36 Common path? In all the cases, Hence Q t f (x) = t s Q s f (x) ds = lim Q tf (x) = t lim Q tf (x) drives to an important operator: t t x P s f (x) ds = x P s f (x) ds (1) lim i sign k r k a k e ikθ = i sign k a k e ikθ (Conjugate function). t k k (2) lim i sign ξe t ξ f (ξ)e iξx dξ = Ĥf (ξ)e iξx dξ (Hilbert transform). t R R t () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

37 Common path? In all the cases, Hence Q t f (x) = t s Q s f (x) ds = lim Q tf (x) = t lim Q tf (x) drives to an important operator: t t x P s f (x) ds = x P s f (x) ds (1) lim i sign k r k a k e ikθ = i sign k a k e ikθ (Conjugate function). t k k (2) lim sign ξe t ξ f (ξ)e iξx dξ = Ĥf (ξ)e iξx dξ (Hilbert transform). i t R (3) lim t t s Q s f (x) ds = x P s f (x) ds R t () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

38 Common path? In all the cases, Hence Q t f (x) = t s Q s f (x) ds = lim Q tf (x) = t lim Q tf (x) drives to an important operator: t t x P s f (x) ds = x P s f (x) ds (1) lim i sign k r k a k e ikθ = i sign k a k e ikθ (Conjugate function). t k k (2) lim sign ξe t ξ f (ξ)e iξx dξ = Ĥf (ξ)e iξx dξ (Hilbert transform). i t R (3) lim t t s Q s f (x) ds = x P s f (x) ds R x P s f (x) ds? () Orthogonal systems and semigroups Logroño, 23 Febrero / 29 t

39 Diffusion semigroups E. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley theory, Princeton, 197. (M, dµ) measure space. {T t } t> : L 2 L 2 : T t1+t 2 = T t1 T t2. T = Id. lim t T t f = f in L 2. T t f p f p, (1 p ). Contraction. T t selfadjoint in L 2. T t f si f. Positivity. T t 1 = 1. Markov. () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

40 First example of semigroup. Classical heat equation The example of diffusion semigroup in L 2 (R): 1 T t f (x) = e x y 2 (4πt) 1/2 4t f (y) dy R () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

41 First example of semigroup. Classical heat equation The example of diffusion semigroup in L 2 (R): 1 T t f (x) = e x y 2 (4πt) 1/2 4t f (y) dy For good functions f we have: t (T t f (x)) = 2 x (T t f (x)) = x (T t f (x)) R In symbols T t f (x) = e t f (x) heat semigroup () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

42 First example of semigroup. Classical heat equation The example of diffusion semigroup in L 2 (R): 1 T t f (x) = e x y 2 (4πt) 1/2 4t f (y) dy For good functions f we have: t (T t f (x)) = 2 x (T t f (x)) = x (T t f (x)) R In symbols Interesting remark T t f (x) = e t f (x) heat semigroup T t f (ξ) = e t ξ 2 f (ξ) () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

43 Second example of diffusion semigroup. Orthogonal polynomials Ilustration. L with eigenfunctions {φ k } k and eigenvalues {λ k } k e tl φ k (x) = e tλ k φ k (x), e tl( ) c k φ k (x) = e tλ k c k φ k (x) k k () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

44 Second example of diffusion semigroup. Orthogonal polynomials Ilustration. L with eigenfunctions {φ k } k and eigenvalues {λ k } k e tl φ k (x) = e tλ k φ k (x), e tl( ) c k φ k (x) = e tλ k c k φ k (x) k k We have the heat equation for L: t (e tl f (x)) = Le tl f (x) () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

45 Second example of diffusion semigroup. Orthogonal polynomials Ilustration. L with eigenfunctions {φ k } k and eigenvalues {λ k } k e tl φ k (x) = e tλ k φ k (x), e tl( ) c k φ k (x) = e tλ k c k φ k (x) k k We have the heat equation for L: t (e tl f (x)) = Le tl f (x) Fourier series case. e ikθ = k 2 e ikθ. e t( )( a k e ikθ) = k k e t k 2 a k e ikθ () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

46 Second example of diffusion semigroup. Orthogonal polynomials Ilustration. L with eigenfunctions {φ k } k and eigenvalues {λ k } k e tl φ k (x) = e tλ k φ k (x), e tl( ) c k φ k (x) = e tλ k c k φ k (x) k k We have the heat equation for L: t (e tl f (x)) = Le tl f (x) Fourier series case. e ikθ = k 2 e ikθ. e t( )( a k e ikθ) = k k e t k 2 a k e ikθ Hermite polynomials case. L = x 2 + 2x x, LH n = 2nH n. Then e tl( ) a n H n (x) = e t2n a n H n (x) n (remember Muckenhoupt) n () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

47 Third example of Poisson semigroup Formula for Gamma function: e t λ = t 2 π e t2 4s s 3/2 e sλ ds, λ >. () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

48 Third example of Poisson semigroup Formula for Gamma function: e t λ = t 2 π e t2 4s s 3/2 e sλ ds, λ >. If L is positive, e t L f = t 2 π e t2 4s s 3/2 e sl f ds. Harmonic ( 2 t L ) (e t L f ) = e t L f : Poisson semigroup () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

49 Third example of Poisson semigroup Formula for Gamma function: e t λ = t 2 π e t2 4s s 3/2 e sλ ds, λ >. If L is positive, e t L f = t 2 π e t2 4s s 3/2 e sl f ds. Harmonic ( 2 t L ) (e t L f ) = e t L f : Poisson semigroup If L can be decompose L = ( x ) ( x ), we get the conjugate harmonic function: Q t f = t s Q s f ds = t x P s f ds () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

50 Determination of kernels If we know e tl then we know e t L. () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

51 Determination of kernels If we know e tl then we know e t L. t 2 π e t2 4s s 3/2 e sl (x, y) ds = t 2 e t2 /4s π = }{{} ( t2 + x y 2 4s =u) = t π s 3/2 1 t π(t 2 + x y 2 ). e x y 2 4s 4πs ds 1 (t 2 + x y 2 ) e u du () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

52 Determination of kernels If we know e tl then we know e t L. t 2 π e t2 4s s 3/2 e sl (x, y) ds = t 2 e t2 /4s π = }{{} ( t2 + x y 2 4s =u) = t π s 3/2 1 t π(t 2 + x y 2 ). e x y 2 4s 4πs ds 1 (t 2 + x y 2 ) e u du Conclusion. If we know the kernel of the heat semigroup, we know the kernel of the Poisson semigroup. () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

53 Riesz Transforms Gamma function firmula: λ α = 1 Γ(α) t α e λt dt, λ, α >. t () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

54 Riesz Transforms Gamma function firmula: λ α = 1 Γ(α) t α e Given φ k, eigenfunction of L with eigenvalue λ k, we have λt dt, λ, α >. t L α φ k = 1 λ α φ k = 1 k Γ(α) t α e λ k t φ k dt t = 1 Γ(α) t α e tl φ k dt t Hence, L α f = 1 Γ(α) t α e tl f dt t () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

55 Riesz Transforms Gamma function firmula: λ α = 1 Γ(α) t α e Given φ k, eigenfunction of L with eigenvalue λ k, we have λt dt, λ, α >. t L α φ k = 1 λ α φ k = 1 k Γ(α) t α e λ k t φ k dt t = 1 Γ(α) t α e tl φ k dt t Hence, L α f = 1 Γ(α) t α e tl f dt t L 1/2 f = ( L) 1 f = 1 Γ(1) Going back to our slide (?) t 1 e t L f dt t = e t L f dt lim Q tf = lim t t t s Q s f (x)ds = x P s f (x)ds = x L 1/2 f () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

56 Stein knew everything!.

57 Stein knew everything!. Look into the red book. 197! () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

58 Stein knew it... () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

59 Boundedness in L 2 of Riesz transforms L = x x () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

60 Boundedness in L 2 of Riesz transforms L = x x Eigenvalue case: ψ k = x (L) 1/2 φ k ( ) ψ k ψ l dµ = x (L) 1/2 φ k )( x (L) 1/2 φ l dµ ( )( ( ) = x x (L) 1/2 φ k (L) 1/2 φ l )dµ = L(L) 1/2 φ k )((L) 1/2 φ l dµ ( ) = (L) 1/2 φ k )((L) 1/2 φ l dµ = λ 1/2 k λ 1/2 l φ k φ l dµ () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

61 Boundedness in L 2 of Riesz transforms L = x x Eigenvalue case: ψ k = x (L) 1/2 φ k ( ) ψ k ψ l dµ = x (L) 1/2 φ k )( x (L) 1/2 φ l dµ ( )( ( ) = x x (L) 1/2 φ k (L) 1/2 φ l )dµ = L(L) 1/2 φ k )((L) 1/2 φ l dµ ( ) = (L) 1/2 φ k )((L) 1/2 φ l dµ = λ 1/2 k λ 1/2 l φ k φ l dµ Spectral theorem: x (L) 1/2 f, x (L) 1/2 f = x x (L) 1/2 f, (L) 1/2 f = (L) 1/2 f, (L) 1/2 f = f, f () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

62 Boundedness in L p of Riesz transforms General procedure in Harmonic Analysis. Once we know the L 2 -boundedness, the kernel is used to get L p boundedness. () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

63 Boundedness in L p of Riesz transforms General procedure in Harmonic Analysis. Once we know the L 2 -boundedness, the kernel is used to get L p boundedness. x L 1/2 f (x) = x e tl 1/2 dt f (x) t t = x e tl 1/2 dt f (x) t t = x e tl 1/2 dt (x, y)f (y) dy t R t ( n ) = x e tl 1/2 dt (x, y) t f (y) dy R n t = K(x, y)f (y) dy. R n () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

64 Possible Applications a priori estimates a priori estimates L = i ( i ) i Lu = f f L p = i j u L p () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

65 Possible Applications a priori estimates a priori estimates L = i ( i ) i Lu = f f L p = i j u L p i j u = i j L 1 f L p () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

66 Application Sobolev Spaces Assume f L p L 1/2 f L p. Equivalently L 1/2 g L p g L p. () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

67 Application Sobolev Spaces Assume f L p L 1/2 f L p. Equivalently Then L 1/2 g L p g L p. {f L p : L 1/2 f L p } = {f L p : f L p } () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

68 Application Sobolev Spaces Assume f L p L 1/2 f L p. Equivalently Then L 1/2 g L p g L p. {f L p : L 1/2 f L p } = {f L p : f L p } Sobolev spaces W k,p L = {f L p : k f L p } = {f L p : L k/2 f L p } () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

69 Muchas gracias! () Orthogonal systems and semigroups Logroño, 23 Febrero / 29

### 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

Διαβάστε περισσότερα

### Uniform Convergence of Fourier Series Michael Taylor

Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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) :=

Διαβάστε περισσότερα

### Problem Set 9 Solutions. θ + 1. θ 2 + cotθ ( ) sinθ e iφ is an eigenfunction of the ˆ L 2 operator. / θ 2. φ 2. sin 2 θ φ 2. ( ) = e iφ. = e iφ cosθ.

Chemistry 362 Dr Jean M Standard Problem Set 9 Solutions The ˆ L 2 operator is defined as Verify that the angular wavefunction Y θ,φ) Also verify that the eigenvalue is given by 2! 2 & L ˆ 2! 2 2 θ 2 +

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### Bessel 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 περισσότερα

### 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

Διαβάστε περισσότερα

### Exercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.

Exercises 0 More exercises are available in Elementary Differential Equations. If you have a problem to solve any of them, feel free to come to office hour. Problem Find a fundamental matrix of the given

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### Boundedness of Some Pseudodifferential Operators on Bessel-Sobolev Space 1

M a t h e m a t i c a B a l k a n i c a New Series Vol. 2, 26, Fasc. 3-4 Boundedness of Some Pseudodifferential Operators on Bessel-Sobolev Space 1 Miloud Assal a, Douadi Drihem b, Madani Moussai b Presented

Διαβάστε περισσότερα

### 2. Let H 1 and H 2 be Hilbert spaces and let T : H 1 H 2 be a bounded linear operator. Prove that [T (H 1 )] = N (T ). (6p)

Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 2005-03-08 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### Mean-Variance Analysis

Mean-Variance Analysis Jan Schneider McCombs School of Business University of Texas at Austin Jan Schneider Mean-Variance Analysis Beta Representation of the Risk Premium risk premium E t [Rt t+τ ] R1

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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.

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### ECE Spring Prof. David R. Jackson ECE Dept. Notes 2

ECE 634 Spring 6 Prof. David R. Jackson ECE Dept. Notes Fields in a Source-Free Region Example: Radiation from an aperture y PEC E t x Aperture Assume the following choice of vector potentials: A F = =

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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.

Διαβάστε περισσότερα

### 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:

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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,

Διαβάστε περισσότερα

### On the Einstein-Euler Equations

On the Einstein-Euler Equations Tetu Makino (Yamaguchi U, Japan) November 10, 2015 / Int l Workshop on the Multi-Phase Flow at Waseda U 1 1 Introduction. Einstein-Euler equations: (A. Einstein, Nov. 25,

Διαβάστε περισσότερα

### 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)

Διαβάστε περισσότερα

### Geodesic Equations for the Wormhole Metric

Geodesic Equations for the Wormhole Metric Dr R Herman Physics & Physical Oceanography, UNCW February 14, 2018 The Wormhole Metric Morris and Thorne wormhole metric: [M S Morris, K S Thorne, Wormholes

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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,

Διαβάστε περισσότερα

### Nonlinear Fourier transform for the conductivity equation. Visibility and Invisibility in Impedance Tomography

Nonlinear Fourier transform for the conductivity equation Visibility and Invisibility in Impedance Tomography Kari Astala University of Helsinki CoE in Analysis and Dynamics Research What is the non linear

Διαβάστε περισσότερα

### 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,

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### Variational Wavefunction for the Helium Atom

Technische Universität Graz Institut für Festkörperphysik Student project Variational Wavefunction for the Helium Atom Molecular and Solid State Physics 53. submitted on: 3. November 9 by: Markus Krammer

Διαβάστε περισσότερα

### The semiclassical Garding inequality

The semiclassical Garding inequality We give a proof of the semiclassical Garding inequality (Theorem 4.1 using as the only black box the Calderon-Vaillancourt Theorem. 1 Anti-Wick quantization For (q,

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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,

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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,

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### Answer sheet: Third Midterm for Math 2339

Answer sheet: Third Midterm for Math 339 November 3, Problem. Calculate the iterated integrals (Simplify as much as possible) (a) e sin(x) dydx y e sin(x) dydx y sin(x) ln y ( cos(x)) ye y dx sin(x)(lne

Διαβάστε περισσότερα

### ( 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

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### Lecture 26: Circular domains

Introductory lecture notes on Partial Differential Equations - c Anthony Peirce. Not to be copied, used, or revised without eplicit written permission from the copyright owner. 1 Lecture 6: Circular domains

Διαβάστε περισσότερα

### 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)

Διαβάστε περισσότερα

### Dispersive estimates for rotating fluids and stably stratified fluids

Διαβάστε περισσότερα

### 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)

Διαβάστε περισσότερα

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

Διαβάστε περισσότερα

### Exercise 1.1. Verify that if we apply GS to the coordinate basis Gauss form ds 2 = E(u, v)du 2 + 2F (u, v)dudv + G(u, v)dv 2

Math 209 Riemannian Geometry Jeongmin Shon Problem. Let M 2 R 3 be embedded surface. Then the induced metric on M 2 is obtained by taking the standard inner product on R 3 and restricting it to the tangent

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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 :

Διαβάστε περισσότερα

### M 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 περισσότερα

### HOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:

HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying

Διαβάστε περισσότερα

### 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.

Διαβάστε περισσότερα

### Appendix A. Curvilinear coordinates. A.1 Lamé coefficients. Consider set of equations. ξ i = ξ i (x 1,x 2,x 3 ), i = 1,2,3

Appendix A Curvilinear coordinates A. Lamé coefficients Consider set of equations ξ i = ξ i x,x 2,x 3, i =,2,3 where ξ,ξ 2,ξ 3 independent, single-valued and continuous x,x 2,x 3 : coordinates of point

Διαβάστε περισσότερα

### 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.

Διαβάστε περισσότερα

### 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,

Διαβάστε περισσότερα

### ( ) 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

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### Concrete Mathematics Exercises from 30 September 2016

Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### Branching form of the resolvent at threshold for discrete Laplacians

Branching form of the resolvent at threshold for discrete Laplacians Kenichi ITO (Kobe University) joint work with Arne JENSEN (Aalborg University) 9 October 2016 Introduction: Discrete Laplacian 1 Thresholds

Διαβάστε περισσότερα

### derivation of the Laplacian from rectangular to spherical coordinates

derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used

Διαβάστε περισσότερα

### Exercises to Statistics of Material Fatigue No. 5

Prof. Dr. Christine Müller Dipl.-Math. Christoph Kustosz Eercises to Statistics of Material Fatigue No. 5 E. 9 (5 a Show, that a Fisher information matri for a two dimensional parameter θ (θ,θ 2 R 2, can

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### The Jordan Form of Complex Tridiagonal Matrices

The Jordan Form of Complex Tridiagonal Matrices Ilse Ipsen North Carolina State University ILAS p.1 Goal Complex tridiagonal matrix α 1 β 1. γ T = 1 α 2........ β n 1 γ n 1 α n Jordan decomposition T =

Διαβάστε περισσότερα

### arxiv: v1 [math-ph] 4 Jun 2016

On commuting ordinary differential operators with polynomial coefficients corresponding to spectral curves of genus two Valentina N. Davletshina, Andrey E. Mironov arxiv:1606.0136v1 [math-ph] Jun 2016

Διαβάστε περισσότερα

### 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.

Διαβάστε περισσότερα

### Computing the Macdonald function for complex orders

Macdonald p. 1/1 Computing the Macdonald function for complex orders Walter Gautschi wxg@cs.purdue.edu Purdue University Macdonald p. 2/1 Integral representation K ν (x) = complex order ν = α + iβ e x

Διαβάστε περισσότερα

### 6.3 Forecasting ARMA processes

122 CHAPTER 6. ARMA MODELS 6.3 Forecasting ARMA processes The purpose of forecasting is to predict future values of a TS based on the data collected to the present. In this section we will discuss a linear

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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) =

Διαβάστε περισσότερα

### Nonlinear Fourier transform and the Beltrami equation. Visibility and Invisibility in Impedance Tomography

Nonlinear Fourier transform and the Beltrami equation Visibility and Invisibility in Impedance Tomography Kari Astala University of Helsinki Beltrami equation: z f(z) = µ(z) z f(z) non linear Fourier transform

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα

### 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,...,

Διαβάστε περισσότερα

### Empirical best prediction under area-level Poisson mixed models

Noname manuscript No. (will be inserted by the editor Empirical best prediction under area-level Poisson mixed models Miguel Boubeta María José Lombardía Domingo Morales eceived: date / Accepted: date

Διαβάστε περισσότερα

### Additional Results for the Pareto/NBD Model

Additional Results for the Pareto/NBD Model Peter S. Fader www.petefader.com Bruce G. S. Hardie www.brucehardie.com January 24 Abstract This note derives expressions for i) the raw moments of the posterior

Διαβάστε περισσότερα

### 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

Διαβάστε περισσότερα