Phase transitions in Kinetic Theory


 Ada McLaughlin
 11 months ago
 Views:
Transcription
1 Phase transitions in Kinetic Theory Brown  FRG Meeting  May 2010 Raffaele Esposito Dipartimento di Matematica pura ed applicata Università dell Aquila, Italy Phase transitions in Kinetic Theory p. 1/48
2 Outline The particle model. Kinetic description. Equilibrium, phase transition. Results: Stability for VB and VFP in R; Instability for VB in R. Joint works with Yan Guo and Rossana Marra: Arch.Rat.Mech.Anal.195, (2010), Commun.Math.Phys.296,133 (2010). Work in progress in a finite interval. Phase transitions in Kinetic Theory p. 2/48
3 The model Model introduced by Bastea and Lebowitz (1997). System ofn 1 red and N 2 blue hard spheres of radius a = 1 with unit mass in Λ R 3. Interactions: elastic collisions (color blind); long range Kactype interaction between red and blue particles with a two body potential: ( ) x φ l (x) = A l U, U(r) = 0 for r > 1. l U is assumed smooth, non negative, decreasing and normalized ( dxu( x ) = 1) and A l 0 as l suitably. Phase transitions in Kinetic Theory p. 3/48
4 BoltzmannGrad+Vlasov limit Λ = O(l 3 ) Mean free path λ = γ = λ l fixed. Λ (N 1 +N 2 )a 2. A l = γ 3 a 2 λ 2 = O(N 1 +N 2 ). Note that N 1 +N 2 = O(l 2 ), low density. By scaling space and time variables (with λ 1 ) and taking the formal limit in the BBGKY hierarchy one gets the kinetic equations below. Warning : The analog of Lanford theorem is not been proved. Phase transitions in Kinetic Theory p. 4/48
5 BoltzmannVlasov equation BV t f 1 +v x f 1 +F 1 [f 2 ] v f 1 = B(f 1,f 1 )+B(f 1,f 2 ), t f 2 +v x f 2 +F 2 [f 1 ] v f 2 = B(f 2,f 2 )+B(f 2,f 1 ), f 1 = f red and f 2 = f blue probability distribution functions on the phase space Ω R 3, Ω = λ 1 Λ; Selfconsistent forces F 1 [f 2 ], F 2 [f 1 ]: F i [f j ](x,t) = x dx γ 3 U(γ x x ) dvf j (x,v,t), i j. R 3 Ω Collisions: B(f,g) = dv R 3 ω =1 dω (v v ) ω [f(v )g(v ) f(v)g(v )], Phase transitions in Kinetic Theory p. 5/48
6 Remark Why not just one species? The interesting case for phase transitions, with one species, is the attractive one: U(r) 0. In order to have thermodynamic stability one needs to keep the hard core size finite on the kinetic scale. Not compatible with the GradBoltzmann limit. Phase transitions in Kinetic Theory p. 6/48
7 VlasovFokkerPlanck eq.n VFP No elastic collisions; system in contact with a thermal bath at inverse temperature β > 0, modeled by white noise force added to the deterministic evolution with long range interaction. Standard mean field arguments: limiting system t f 1 +v x f 1 +F 1 [f 2 ] v f 1 = Lf 1, t f 2 +v x f 2 +F 2 [f 1 ] v f 2 = Lf 2, Lf(v) = v (µ β v ( fµβ )), µ β = exp[ βv2 ] (2πβ 1 ) 3/2. Phase transitions in Kinetic Theory p. 7/48
8 Segregation Phase transitions in Kinetic Theory p. 8/48
9 Free energy functional F kin Ω (f1,f 2 ) = E Ω (f 1,f 2 )+β 1 H Ω (f 1,f 2 ), E Ω (f 1,f 2 ) = dx dv 1 [ ] Ω R 2 v2 f 1 (x,v)+f 2 (x,v) 3 + dx dx U( x x )ρ 1 (x)ρ 2 (x ), Ω Ω ρ i (x) = H Ω (f 1,f 2 ) = i=1,2 Ω dvf i (x,v), R 3 dx dvf i (x,v)lnf i (x,v). R 3 Phase transitions in Kinetic Theory p. 9/48
10 The functional FΩ kin is non increasing for any β > 0 under the VB dynamics and for β given by the thermal bath inverse temperature for the VFP evolution. Entropyenergy competition. β small: entropy dominates. Disordered state. β large: energy dominates. Segregated states. Phase transitions in Kinetic Theory p. 10/48
11 Equilibrium Equilibrium solutions (both VB and VFP): f 1 (x,v) = ρ 1 (x)µ β (v), β 1 lnρ 1 (x)+ β 1 lnρ 2 (x)+ Ω Ω f 2 (x,v) = ρ 2 (x)µ β (v), dx U( x x )ρ 2 (x ) = C 1, dx U( x x )ρ 1 (x ) = C 2. EulerLagrange equations for the spatial free energy functional F Ω [ρ 1,ρ 2 ] = β 1 dx[ρ 1 (x)lnρ 1 (x)+ρ 2 (x)lnρ 2 (x)] Ω + dx dx U( x x )ρ 1 (x)ρ 2 (x ). Ω Ω Phase transitions in Kinetic Theory p. 11/48
12 Local free energy Homogeneous minimizers: minimizers of the local free energy ϕ(ρ 1,ρ 2 ) = β 1 [ρ 1 lnρ 1 +ρ 2 lnρ 2 ]+ρ 1 ρ 2 Indeed, Ω dx F Ω [ρ 1,ρ 2 ] = dxϕ(ρ 1 (x),ρ 2 (x)) Ω dx U( x x )(ρ 1 (x) ρ 1 (x ))(ρ 2 (x ) ρ 2 (x)). Ω By rearrangement arguments (U decreasing) the non local term is non negative on minimizers, so minimizers are spatially homogeneous as much as they can. Phase transitions in Kinetic Theory p. 12/48
13 Homogeneous equilibrium states Set ρ = ρ 1 +ρ 2 ; m = ρ 1 ρ 2 ρ. ϕ(ρ 1,ρ 2 ) = β 1 ρlnρ+ 1 4 ρ2 +ρ 2 f ρ (m) f ρ (m) = m2 4 +(ρβ) 1[ 1 m 2 ln 1 m m 2 ln 1+m ] 2 Graph ofm f ρ (m) forρβ > 2. Phase transitions in Kinetic Theory p. 13/48
14 Homogeneous equilibrium states The critical points for f have to solve with σ = 1 2 ρβ. m = tanh(σm), If σ 1: m = 0 is the only solution. If σ > 1: again m = 0 solution; Moreover m σ > 0 such that m = ±m σ is solution. Set ρ ± := 1 2 ρ(1±m1 2 ρβ). Phase transitions in Kinetic Theory p. 14/48
15 Homogeneous equilibrium states Therefore: ρβ 2 : m = 0: Miminizer of ϕ: ρ 1 = ρ 2 (mixed phase). ρβ > 2 : Minimizer: ρ 1 = ρ +,ρ 2 = ρ,(red rich phase); Minimizer: ρ 1 = ρ,ρ 2 = ρ +,(blue rich phase); Maximizer (local) : ρ 1 = ρ 2. Note: the minimizing total density is always unique. Phase transitions in Kinetic Theory p. 15/48
16 Non homogeneous equilibrium states Conservation of masses = Minimizers with the constraints 1 dxρ i (x) = n i, i = 1,2 Ω Ω n i given by the initial conditions. If < ρ n i < ρ +, i = 1,2 non homogeneous minimizers, regions blue rich and red rich separated by interfaces. Assume Ω a torus of size L. From rearrangement arguments (using U monotone), for L sufficiently large, the minimizers are symmetric monotone with values close to ρ + and ρ in regions separated by a small interface [Carlen, Carvalho, R.E., Lebowitz, Marra, Nonlinearity 16, (2003)]. Phase transitions in Kinetic Theory p. 16/48
17 Equilibrium states in R Ω = R. Fix ρ = 2 = σ = β; Critical point β c = 1. For β 1: the only possible equilibrium is the homogeneous mixed phase, with ρ 1 = ρ 2 = 1. For β > 1: three homogeneous states: the mixed one, ρ 1 = ρ 2, the red rich phase ρ 1 = 1+m β, ρ 2 = 1 m β and the blue rich phase ρ 1 = 1 m β, ρ 2 = 1+m β. For β > 1, non spatially homogeneous solutions (phase coexistence) by forcing the asymptotic values to ρ ± at : Two halflines of red rich and blue rich phases separated by an interface of finite size. Phase transitions in Kinetic Theory p. 17/48
18 Equilibrium states in R Define ˆρ 1 (x) = { ρ, x < 0 ρ +, x > 0, ˆρ 2(x) = { ρ +, x < 0 ρ, x > 0. Excess free energy: [ ] F[ρ 1,ρ 2 ] = lim F ( l,l) [ρ 1,ρ 2 ] F ( l,l) [ˆρ 1, ˆρ 2 ]. l Remark: F[ρ 1,ρ 2 ] is not finite if lim ρ 1 ρ ± or x ± lim ρ 2 ρ. x ± Phase transitions in Kinetic Theory p. 18/48
19 Front solution Theorem[Carlen, Carvalho, R.E., Lebowitz, Marra; ARMA 194, (2009)] Letβ > 1. There is a unique (up to translations) minimizer to the excess free energy F. Let ρ = ( ρ1, ρ 2 ) be the one such that ρ 1 (0) = ρ 2 (0). ρ is smooth;ρ < ρ i (x) < ρ +, i=1,2; ρ 1 is increasing and ρ 2 is decreasing; β 1 ln ρ 1 +U ρ 2 = C = β 1 ln ρ 2 +U ρ 1 ; Phase transitions in Kinetic Theory p. 19/48
20 Front solution Moreover β 1 ρ 1 + ρ 1U ρ 2 = 0 = β 1 ρ 2 + ρ 1U ρ 1 ; ρ 1 (x) = ρ 2 ( x), ρ 1 (x) = ρ 2 ( x); α > 0: ρ 1 (x) ρ ± e α x 0, x ± ; ρ 2 (x) ρ e α x 0, x ±. Phase transitions in Kinetic Theory p. 20/48
21 Front solution Phase transitions in Kinetic Theory p. 21/48
22 Stability Question: Are the equilibrium states stable w.r.t. perturbations of the initial conditions in the time evolutions VB and VFP? Expected answer: Minimizers are stable, maximizers are unstable. Results: Case Ω = R; VFP evolution: asymptotic stability of the minimizers. VB evolution: stability of the minimizers, instability of the maximizer. Open problems: Instability of the maximizer for VFP and asymptotic stability of the minimizers for VB. Ω = [ L,L], periodic b.c.,llarge enough. Phase transitions in Kinetic Theory p. 22/48
23 Stability for the VFP Main problem: the force cannot be small: small force = no phase transition. For small force, convergence to equilibrium and rate by Villani. This would cover the high temperature regime. For low temperatures, multiple equilibrium states. Difficult case: front. Homogeneous minimizers are simpler. Assume f i 0 (x,v) = ρ i(x)µ β (v)+h i (x,v,0), x R, v R 3. Define h i (x,v,t) := f i (x,v,t) ρ i (x)µ β (v). Phase transitions in Kinetic Theory p. 23/48
24 Stability for the VFP Equations for h i : t h i +G i h i = Lh i F i [h] vx h i, G i h i = v x x h i (U ρ j) vx h i ) + (U x dvh j (,v,t) βv x M ρ i R 3 Norms: denotes the L 2 (R R 3 ) with weight ( ρ i µ β ) 1 : f 2 = i=1,2 R R3 dxdv f(x,v)2 ρ(x)µ β (v) Phase transitions in Kinetic Theory p. 24/48
25 Stability for the VFP Null space of L: kern(l) = {(u 1,u 2 ) = (a 1 µ β,a 2 µ β )}. P projector on kern(l),p = 1 P ; = ( x, t ). Dissipation norm: f 2 D = P f 2 + v P f 2. Weighted norms: z γ = (1+ x 2 ) γ, f γ = z γ f ; f D,γ = z γ f D Symmetry: Assume h 1 (x,v,0) = h 2 ( x,rv,0) with Rv = ( v x,v y,v z ). Note that this is the same symmetry of the front. The symmetry is preserved in time. Phase transitions in Kinetic Theory p. 25/48
26 Stability for the VFP Theorem[ARMA 195, (2010)] There is δ > 0 s.t. if h(,0) + h(,0) < δ, then there is a unique global solution to the equation forh. Moreover there isk > 0 such that d ( h(t) 2 + t h(t) 2 + x h(t) 2) dt +K ( h(t) 2 D + th(t) 2 D + xh(t) 2 D) 0. If, forγ > 0 sufficiently small, h(,0) γ + h(,0) γ+ 1 2 < δ then the same norms are bounded at any time by the initial data and we have the decay estimate h(t) 2 + h(t) 2 C [ 1+ t ] [ ] 2γ h(0) 2 γ + h(0) 2 1 2γ +γ. 2 Phase transitions in Kinetic Theory p. 26/48
27 Main tools Spectral gap for L: there is ν 0 > 0 s.t. (g,lg) ν 0 P g 2 D. This is used to control P h(t). Control P h(t) = (g 1 µ β,g 2 µ β ): (Ag) 1 = β 1g 1 ρ 1 +U g 2, (Ag) 2 = β 1g 2 ρ 2 +U g 1. Note that (g,ag) = d2 ds 2 ˆF( ρ+sg) s=0. Phase transitions in Kinetic Theory p. 27/48
28 Main tools The operator A has a null space kern(a) = {α( ρ 1, ρ 2 )}. Let P is the projector on kern(a) and P = 1 P. (Transl. invar.) Spectral gap for A: there is λ > 0 s.t. (g,ag) λ(p g,p g). The component of h(t) in kern(a) is not under control. The symmetry condition ensures that h(t) is orthogonal to kern(a). Lower bound for (Au) : There is C > 0 such that (Au) 2 C Qu 2. where Q is the orthogonal projection on the orthogonal complement of ρ. Phase transitions in Kinetic Theory p. 28/48
29 Remarks The same results hold for the mixed phase above the critical temperature. We do not have the instability of the mixed phase below the critical temperature. The construction of the growing mode presented below for VB, fails for VFP because of the unboundedness of the FP operator L. Phase transitions in Kinetic Theory p. 29/48
30 Results for VB Theorem[CMP 296,133 (2010)]: Assume ρ = 2. β < 1: The unique equilibrium(f 1,f 2 ) = (µ β,µ β ) is stable. β > 1: the homogeneous equilibrium states (f 1,f 2 ) = (ρ + µ β,ρ µ β ) and(f 1,f 2 ) = (ρ µ β,ρ + µ β ) are stable; the equilibrium(f 1,f 2 ) = ( ρ 1 (x)µ β, ρ 2 (x)µ β ) is stable w.r.t. symmetric perturbations; the homogeneous equilibrium(f 1,f 2 ) = (µ β,µ β ) is unstable. Here stability and instability are in L (R R 3 ) and in H 1 (R R 3 ). Symmetric perturbation means again h 1 (x,v) = h 2 ( x,rv), where Rv = ( v x,v y,v z ). Phase transitions in Kinetic Theory p. 30/48
31 Remarks No convergence to the equilibrium is stated. This has to be compared with the VlasovFokkerPlank case where there is an algebraic rate of convergence. No instability result for VFP. In order to have phase transitions: force not small. Treating the force terms as perturbations does not work. Strategy based on entropyenergy arguments: L 2 estimates promoted to L by analysis of the characteristics. Crucial step: spectral gap for the second derivative of the free energy. The instability is based on the construction of a growing mode for the linear collisionless case, perturbation arguments and persistence of the gorwing mode at non linear level. Phase transitions in Kinetic Theory p. 31/48
32 Free energy functional Given the equilibrium state (M 1,M 2 ) = (ρ 1 µ β,ρ 2 µ β ), let g = (g 1,g 2 ) with g i = f i M i Mi be the deviation from the equilibrium. Define: M i (g) = R dx dv M R 3 i g i (x,v), 2 ] H(g) = dx dv [f i logf i M i logm i, + R R i=1 E(g) = R 2 i=1 R dx dv v2 2 g i Mi ( ) dxdyu( x y ) ρ f1 (x)ρ f2 (y) ρ 1 (x)ρ 2 (y), ρ fi = dvf i (x,v). Phase transitions in Kinetic Theory p. 32/48
33 Free energy functional The free energy functional is ( F(g) = H(g)+βE(g) C +1+log ( ) β 3/2 2 M i (g), 2π) The free energy functional does not increase: F(g(t)) F(g(0)) for any t > 0. Quadratic approximation. The coefficients have been chosen to cancel the linear part. For some f i = αf i +(1 α)m i, α (0,1): 2 R3 F(g) = dx dv (f i(t) M i ) 2 i=1 R 2 f i +β dx dyu( x y )(ρ f1 (t,x) ρ 1 (x))(ρ f2 (t,y) ρ 2 (y)). R R i=1 Phase transitions in Kinetic Theory p. 33/48
34 Free energy functional Lemma. Ifu i = ρ fi ρ i s.t. P(u 1,u 2 ) = 0, then there areα > 0 and κ sufficiently small so that α i=1,2 R { (f i (t) M i ) 2 dx dv 1 R M { fi (t) M i κm i } 3 i + f i (t) M i 1 { fi (t) M i κm i } F(g(0)). } Remark: It is crucial that the initial perturbation is orthogonal to the null space ofa. This is trivial for the spatially homogeneous equilibrium, while it is ensured by the symmetry condition for the phase coexisting equilibrium. Phase transitions in Kinetic Theory p. 34/48
35 Main proposition Theorem: Letw(v) = (Σ+ v 2 ) γ, withσsufficiently large and γ > 3 2. If wg(0) + F(g(0)) < δ forδ sufficiently small, then there ist 0 > such that wg(t 0 ) 1 2 wg(0) +C T0 F(g(0)). The stability follows by iteration on the time interval. Phase transitions in Kinetic Theory p. 35/48
36 Growing mode Idea: Without collisions there is a growing mode. Collisions do not destroy the growing mode. The linearization of the equation for the perturbation is: t g +Lg = 0, Notation : µ = µ β ;ξ first component of the velocity v = (ξ,ζ), (Lg) i = ξ x g i βf( µg i+1 )ξ µ αl i g, α = 1 and L i g = 1 (B( µg i,2µ)+b(µ, ) µ(g 1 +g 2 )). µ Seek for a growing mode of the form g 1 (t,x,ξ,ζ) = e λt e ikx q(ξ,ζ), g 2 (t,x,ξ,ζ) = e λt e ikx q( ξ,ζ). Phase transitions in Kinetic Theory p. 36/48
37 Eigenvalue problem {λ+iξk}q βkiû(k) { R 3 q µdv } ξ µ αlq = 0. Proposition 1: Letβ > 1. There exists sufficiently smallα > 0 such that there is an eigenfunctionq(v) and the eigenvalueλwithrλ > 0. Proof. First assume α = 0. λ is found by O. Penrose criterion, β ξ 2 Û(k)k 2 µ(v) R λ 2 +k 2 ξ 2 dv = 1. 3 Indeed, since βû(0) > 1, by continuity there is k 0 > 0 such that βû(k 0) > 1 and hence a λ > 0 so that this is satisfied. Use Kato perturbation theorem to extend to α > 0 small. Phase transitions in Kinetic Theory p. 37/48
38 Eigenvalue problem Proposition 2: Letα 0 be the supremum of theα s such that Proposition 1 is true. Thenα 0 = +. Proof. Indeed, if α 0 < then, λ 0 = lim α α0 λ α exists (up to subsequences) and is a purely imaginary eigenvalue. It can be shown that the corresponding eigenfunction must be in the null space of L and this implies λ = 0. Moreover, collisions disappear for such an eigenfunction and we can use again the Penrose criterion which implies βû(k 0) = 1. This is in contradiction with the definition of k 0. This provides a linear growing mode for any α > 0. Remark: It is crucial that L is a bounded perturbation. It does not work with the FokkerPlank operator which is unbounded. Non linear analysis. Bootstrap argument. Very technical. Phase transitions in Kinetic Theory p. 38/48
39 Instability theorem Theorem. Assumeβ > 1. There exist constantsk 0 > 0,θ > 0, C > 0,c > 0 and a family of initial 2π k 0 periodic data f δ i (0) = µ+ µg δ i (0) 0, withgδ (0) satisfying x,ξ g δ (0) L 2 + wg δ (0) L Cδ, forδ sufficiently small, but the solutiong δ (t) satisfies sup wg δ (t) L c sup g δ (t) L 2 cθ > 0. 0 t T δ 0 t T δ Here the escape time ist δ = 1 Rλ ln θ δ, Note that the growing mode is symmetric. The instability does not depend on the absence of symmetry. Phase transitions in Kinetic Theory p. 39/48
40 Work in progress Finite volume, 1d torus (with Y. Guo and R. Marra) Stability of the non constant minimizer double front ) Operator A on L 2 (T L ),T L the 1d torus of size L. Derivative of the front is in the null. Null space? Spectral gap? Phase transitions in Kinetic Theory p. 40/48
41 Spectral gap A has a negative eigenvalue. Indeed, let w = (w 1,w 2 ) be a front on the torus such that Aw = 0. Define w = ( w 1, w 2 ) ( w,a w) w = T L w 1 ( w 1 w 1 U w 2 )+ w 2 ( w 2 w 2 U w 1 ) = 2 L 0 w 1U ( w 2 +w 2) 2 We have used the EL equations 0 L w 2U ( w 1 +w 1) < 0 w 1 w 1 = U w 2, x 0; w 2 w 2 = U w 1, x 0 Show that the mass constraint kills the negative eigenvalue. Phase transitions in Kinetic Theory p. 41/48
42 Neumann b.c. Spectral gap true for antisymmetric (by reflection) functions. Problem on the torus for symmetric functions reduced to the case of Neumann boundary conditions on [0,L]: (Âg) 1 = β 1 g 1 w 1 +Û g 2, (Âg) 2 = β 1 g 2 w 2 +βû g 1 Û(z,z ) = U(z,z )+U(z,R 0 z )+U(z,R L z ) R 0 reflection around zero and R L reflection around L. λ L < 0 minimum eigenvalue and ê its eigenfunction. Phase transitions in Kinetic Theory p. 42/48
43 Spectral gap We need spectral gap for functions in the hyperplane H = {h : L 0 h = 0}. We have spectral gap for functions in the orthogonal to ê. (u,âu) δ(u,u), if (u,ê) = 0. If the angle α between ê and H is too small we are in trouble: Phase transitions in Kinetic Theory p. 43/48
44 Spectral gap Decompose h H as aê + bû with û orthogonal to ê (h,âh) = a2 λ L +b 2 (û,âû) a 2 = cos 2 α, b 2 = sin 2 α. with sin α = 1 L L 0 ê. If ê decays fast enough b 2 L 1. λ L is negative Competition (h,âh) λ L + c L δ If λ L decays faster than L 1 (h,âh) > d(h,h) G. Manzi (2007) we can prove spectral gap for L large Phase transitions in Kinetic Theory p. 44/48
45 Spectrum Analysis of the spectrum using Markov chains. Generalize method by De Masi, Olivieri, Presutti (1998), Ising case. bound on the minimum eigenvalue λ L c 1 e γl λ L c 2 e γl exponential bound on the minimum eigenfunction ê ce γ L x ê < 0 spectral gap in a suitable weighted L for functions u in the orthogonal to ê. Implies spectral gap in L 2. Phase transitions in Kinetic Theory p. 45/48
46 Spectral gap Operator S (Su) i = w i Û u j, i j,i = 1,2 (u,âu) = dxw i u i (Âu) i = (u,u)+(u,su) i (u,s 2 u) = i u i w i Û (w j Û u i ) = (u,tu) Negative eigenvalue for Â means eigenvalue for S greater than 1. We study the operators T 1 h = w 1 Û (w 2 Û h), T 2 h = w 2 Û (w 1 Û h). Phase transitions in Kinetic Theory p. 46/48
47 Markov chain Ĵ(x,x ) = dzû(x z)w 2(z) w 2 (x) dyû(z x ) M(x,x ) = p(x)ĵ(x,x ); p(x) = w 1 (x)w 2 (x) For λ L > 0 and ê(x) positive, define the Markov kernel K(x,y) = M(x,y)ê(y) λ 0 ê(x). Phase transitions in Kinetic Theory p. 47/48
48 The End Thanks! Phase transitions in Kinetic Theory p. 48/48
Lecture Notes on PDEs
Lecture Notes on PDEs Alberto Bressan February 26, 2012 1 Elliptic equations Let IR n be a bounded open set Given measurable functions a ij, b i, c : IR, consider the linear, second order differential
More informationPhase Transitions, Hysteresis, and Hyperbolicity for SelfOrganized Alignment Dynamics
Arch. Rational ech. Anal. 6 (05) 63 5 Digital Object Identifier (DOI) 0.007/s00050408007 Phase Transitions, Hysteresis, and Hyperbolicity for SelfOrganized Alignment Dynamics Pierre Degond, Amic Frouvelle
More informationAn introduction to Birkhoff normal form
An introduction to Birkhoff normal form Dario Bambusi Dipartimento di Matematica, Universitá di Milano via Saldini 50, 0133 Milano (Italy) 19.11.14 1 Introduction The aim of this note is to present an
More informationAlignment processes on the sphere
Alignment processes on the sphere Amic Frouvelle CEREMADE Université Paris Dauphine Joint works with : Pierre Degond (Imperial College London) and Gaël Raoul (École Polytechnique) JianGuo Liu (Duke University)
More informationStatistical Mechanics of Active Matter
Statistical Mechanics of Active Matter Umberto Marini Bettolo Marconi University of Camerino, Italy Naples, 24 May,2017 Umberto Marini Bettolo Marconi (2017) Statistical Mechanics of Active Matter 2017
More information1 Lyapunov theory of stability
M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, 29 1 1 Lyapunov theory of stability Introduction. Lyapunov s second (or direct) method provides tools for studying (asymptotic) stability
More informationClassical Density Functional Theory
Classical Density Functional Theory Towards interfaces in crystalline materials Gunnar Pruessner and Adrian Sutton Department of Physics Imperial College London INCEMS M12 meeting, Imperial College London,
More informationTwoBody Problem. Central Potential. 1D Motion
TwoBody Problem. Central Potential. D Motion The simplest nontrivial dynamical problem is the problem of two particles. The equations of motion read. m r = F 2, () We already know that the center of
More informationLecture 12: Detailed balance and Eigenfunction methods
Miranda HolmesCerfon Applied Stochastic Analysis, Spring 2015 Lecture 12: Detailed balance and Eigenfunction methods Readings Recommended: Pavliotis [2014] 4.54.7 (eigenfunction methods and reversibility),
More informationPhase transitions, hysteresis, and hyperbolicity for selforganized alignment dynamics
Phase transitions, hysteresis, and hyperbolicity for selforganized alignment dynamics Pierre Degond (,), Amic Frouvelle (3), JianGuo Liu (4)  Université de Toulouse; UPS, INSA, UT, UT; Institut de athématiques
More informationLecture 15 Newton Method and SelfConcordance. October 23, 2008
Newton Method and SelfConcordance October 23, 2008 Outline Lecture 15 Selfconcordance Notion Selfconcordant Functions Operations Preserving Selfconcordance Properties of Selfconcordant Functions Implications
More informationData Analysis and Manifold Learning Lecture 7: Spectral Clustering
Data Analysis and Manifold Learning Lecture 7: Spectral Clustering Radu Horaud INRIA Grenoble RhoneAlpes, France Radu.Horaud@inrialpes.fr http://perception.inrialpes.fr/ Outline of Lecture 7 What is spectral
More informationBrownian Motion: FokkerPlanck Equation
Chapter 7 Brownian Motion: FokkerPlanck Equation The FokkerPlanck equation is the equation governing the time evolution of the probability density of the Brownian particla. It is a second order differential
More informationRegularization and Inverse Problems
Regularization and Inverse Problems Caroline Sieger Host Institution: Universität Bremen Home Institution: Clemson University August 5, 2009 Caroline Sieger (Bremen and Clemson) Regularization and Inverse
More informationMath 312 Lecture Notes Linear Twodimensional Systems of Differential Equations
Math 2 Lecture Notes Linear Twodimensional Systems of Differential Equations Warren Weckesser Department of Mathematics Colgate University February 2005 In these notes, we consider the linear system of
More informationIntroduction to Selfnormalized Limit Theory
Introduction to Selfnormalized Limit Theory QiMan Shao The Chinese University of Hong Kong Email: qmshao@cuhk.edu.hk Outline What is the selfnormalization? Why? Classical limit theorems Selfnormalized
More informationDerivation of Kinetic Equations
CHAPTER 2 Derivation of Kinetic Equations As we said, the mathematical object that we consider in Kinetic Theory is the distribution function 0 apple f(t, x, v). We will now be a bit more precise about
More informationStability Analysis of Stationary Solutions for the Cahn Hilliard Equation
Stability Analysis of Stationary Solutions for the Cahn Hilliard Equation Peter Howard, Texas A&M University University of Louisville, Oct. 19, 2007 References d = 1: Commun. Math. Phys. 269 (2007) 765
More informationZero Variance Markov Chain Monte Carlo for Bayesian Estimators
Noname manuscript No. will be inserted by the editor Zero Variance Markov Chain Monte Carlo for Bayesian Estimators Supplementary material Antonietta Mira Reza Solgi Daniele Imparato Received: date / Accepted:
More informationCluster Expansion in the Canonical Ensemble
Università di Roma Tre September 2, 2011 (joint work with Dimitrios Tsagkarogiannis) Background Motivation: phase transitions for a system of particles in the continuum interacting via a Kac potential,
More informationUniform exponential decay of the free energy for Voronoi finite volume discretized reactiondiffusion systems
Weierstrass Institute for Applied Analysis and Stochastics Special Session 36 Reaction Diffusion Systems 8th AIMS Conference on Dynamical Systems, Differential Equations & Applications Dresden University
More informationMATHS 730 FC Lecture Notes March 5, Introduction
1 INTRODUCTION MATHS 730 FC Lecture Notes March 5, 2014 1 Introduction Definition. If A, B are sets and there exists a bijection A B, they have the same cardinality, which we write as A, #A. If there exists
More information1 Stochastic Dynamic Programming
1 Stochastic Dynamic Programming Formally, a stochastic dynamic program has the same components as a deterministic one; the only modification is to the state transition equation. When events in the future
More informationGradient Flows: Qualitative Properties & Numerical Schemes
Gradient Flows: Qualitative Properties & Numerical Schemes J. A. Carrillo Imperial College London RICAM, December 2014 Outline 1 Gradient Flows Models Gradient flows Evolving diffeomorphisms 2 Numerical
More informationTopological vectorspaces
(July 25, 2011) Topological vectorspaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ Natural nonfréchet spaces Topological vector spaces Quotients and linear maps More topological
More informationKinetic/Fluid micromacro numerical scheme for VlasovPoissonBGK equation using particles
Kinetic/Fluid micromacro numerical scheme for VlasovPoissonBGK equation using particles Anaïs Crestetto 1, Nicolas Crouseilles 2 and Mohammed Lemou 3. The 8th International Conference on Computational
More informationModern Discrete Probability Spectral Techniques
Modern Discrete Probability VI  Spectral Techniques Background Sébastien Roch UW Madison Mathematics December 22, 2014 1 Review 2 3 4 Mixing time I Theorem (Convergence to stationarity) Consider a finite
More informationRelaxation methods and finite element schemes for the equations of viscoelastodynamics. Chiara Simeoni
Relaxation methods and finite element schemes for the equations of viscoelastodynamics Chiara Simeoni Department of Information Engineering, Computer Science and Mathematics University of L Aquila (Italy)
More informationRobotics. Control Theory. Marc Toussaint U Stuttgart
Robotics Control Theory Topics in control theory, optimal control, HJB equation, infinite horizon case, LinearQuadratic optimal control, Riccati equations (differential, algebraic, discretetime), controllability,
More informationLucio Demeio Dipartimento di Ingegneria Industriale e delle Scienze Matematiche
Scuola di Dottorato THE WAVE EQUATION Lucio Demeio Dipartimento di Ingegneria Industriale e delle Scienze Matematiche Lucio Demeio  DIISM wave equation 1 / 44 1 The Vibrating String Equation 2 Second
More informationIntertwinings for Markov processes
Intertwinings for Markov processes Aldéric Joulin  University of Toulouse Joint work with : Michel Bonnefont  Univ. Bordeaux Workshop 2 Piecewise Deterministic Markov Processes ennes  May 1517, 2013
More information2.2. OPERATOR ALGEBRA 19. If S is a subset of E, then the set
2.2. OPERATOR ALGEBRA 19 2.2 Operator Algebra 2.2.1 Algebra of Operators on a Vector Space A linear operator on a vector space E is a mapping L : E E satisfying the condition u, v E, a R, L(u + v) = L(u)
More informationEDP with strong anisotropy : transport, heat, waves equations
EDP with strong anisotropy : transport, heat, waves equations Mihaï BOSTAN University of AixMarseille, FRANCE mihai.bostan@univamu.fr Nachos team INRIA Sophia Antipolis, 3/07/2017 Main goals Effective
More informationStabilization of persistently excited linear systems
Stabilization of persistently excited linear systems Yacine Chitour Laboratoire des signaux et systèmes & Université ParisSud, Orsay Exposé LJLL Paris, 28/9/2012 Stabilization & intermittent control Consider
More informationarxiv:math/ v1 [math.ap] 28 Oct 2005
arxiv:math/050643v [math.ap] 28 Oct 2005 A remark on asymptotic completeness for the critical nonlinear KleinGordon equation Hans Lindblad and Avy Soffer University of California at San Diego and Rutgers
More informationQuasineutral limit for EulerPoisson system in the presence of plasma sheaths
in the presence of plasma sheaths Department of Mathematical Sciences Ulsan National Institute of Science and Technology (UNIST) joint work with Masahiro Suzuki (Nagoya) and ChangYeol Jung (Ulsan) The
More informationDISSIPATIVE TRANSPORT APERIODIC SOLIDS KUBO S FORMULA. and. Jean BELLISSARD 1 2. Collaborations:
Vienna February 1st 2005 1 DISSIPATIVE TRANSPORT and KUBO S FORMULA in APERIODIC SOLIDS Jean BELLISSARD 1 2 Georgia Institute of Technology, Atlanta, & Institut Universitaire de France Collaborations:
More informationEntanglement Entropy in Extended Quantum Systems
Entanglement Entropy in Extended Quantum Systems John Cardy University of Oxford STATPHYS 23 Genoa Outline A. Universal properties of entanglement entropy near quantum critical points B. Behaviour of entanglement
More informationON MARKOV AND KOLMOGOROV MATRICES AND THEIR RELATIONSHIP WITH ANALYTIC OPERATORS. N. Katilova (Received February 2004)
NEW ZEALAND JOURNAL OF MATHEMATICS Volume 34 (2005), 43 60 ON MARKOV AND KOLMOGOROV MATRICES AND THEIR RELATIONSHIP WITH ANALYTIC OPERATORS N. Katilova (Received February 2004) Abstract. In this article,
More informationMultisolitons for NLS
Multisolitons for NLS Stefan LE COZ Beijing 20070703 Plan 1 Introduction 2 Existence of multisolitons 3 (In)stability NLS (NLS) { iut + u + g( u 2 )u = 0 u t=0 = u 0 u : R t R d x C (A0) (regular) g
More informationVIII. Phase Transformations. Lecture 38: Nucleation and Spinodal Decomposition
VIII. Phase Transformations Lecture 38: Nucleation and Spinodal Decomposition MIT Student In this lecture we will study the onset of phase transformation for phases that differ only in their equilibrium
More information3 Stability and Lyapunov Functions
CDS140a Nonlinear Systems: Local Theory 02/01/2011 3 Stability and Lyapunov Functions 3.1 Lyapunov Stability Denition: An equilibrium point x 0 of (1) is stable if for all ɛ > 0, there exists a δ > 0 such
More informationSalmon: Lectures on partial differential equations
6. The wave equation Of the 3 basic equations derived in the previous section, we have already discussed the heat equation, (1) θ t = κθ xx + Q( x,t). In this section we discuss the wave equation, () θ
More informationBrownian motion. Samy Tindel. Purdue University. Probability Theory 2  MA 539
Brownian motion Samy Tindel Purdue University Probability Theory 2  MA 539 Mostly taken from Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve Samy T. Brownian motion Probability Theory
More informationUnderstanding Regressions with Observations Collected at High Frequency over Long Span
Understanding Regressions with Observations Collected at High Frequency over Long Span Yoosoon Chang Department of Economics, Indiana University Joon Y. Park Department of Economics, Indiana University
More informationNonasymptotic Analysis of Bandlimited Functions. Andrei Osipov Research Report YALEU/DCS/TR1449 Yale University January 12, 2012
Prolate spheroidal wave functions PSWFs play an important role in various areas, from physics e.g. wave phenomena, fluid dynamics to engineering e.g. signal processing, filter design. Even though the significance
More informationAn Observation on the Positive Real Lemma
Journal of Mathematical Analysis and Applications 255, 48 49 (21) doi:1.16/jmaa.2.7241, available online at http://www.idealibrary.com on An Observation on the Positive Real Lemma Luciano Pandolfi Dipartimento
More informationA Toy Model. Viscosity
A Toy Model for Sponsoring Viscosity Jean BELLISSARD Westfälische WilhelmsUniversität, Münster Department of Mathematics SFB 878, Münster, Germany Georgia Institute of Technology, Atlanta School of Mathematics
More informationSmoluchowski Diffusion Equation
Chapter 4 Smoluchowski Diffusion Equation Contents 4. Derivation of the Smoluchoswki Diffusion Equation for Potential Fields 64 4.2 OneDimensionalDiffusoninaLinearPotential... 67 4.2. Diffusion in an
More informationOne Dimensional Dynamical Systems
16 CHAPTER 2 One Dimensional Dynamical Systems We begin by analyzing some dynamical systems with onedimensional phase spaces, and in particular their bifurcations. All equations in this Chapter are scalar
More informationQuasiNewton methods for minimization
QuasiNewton methods for minimization Lectures for PHD course on Numerical optimization Enrico Bertolazzi DIMS Universitá di Trento November 21 December 14, 2011 QuasiNewton methods for minimization 1
More informationKernel Method: Data Analysis with Positive Definite Kernels
Kernel Method: Data Analysis with Positive Definite Kernels 2. Positive Definite Kernel and Reproducing Kernel Hilbert Space Kenji Fukumizu The Institute of Statistical Mathematics. Graduate University
More informationEXISTENCE AND UNIQUENESS FOR A THREE DIMENSIONAL MODEL OF FERROMAGNETISM
1 EXISTENCE AND UNIQUENESS FOR A THREE DIMENSIONAL MODEL OF FERROMAGNETISM V. BERTI and M. FABRIZIO Dipartimento di Matematica, Università degli Studi di Bologna, P.zza di Porta S. Donato 5, I4126, Bologna,
More informationLecture 17 Brownian motion as a Markov process
Lecture 17: Brownian motion as a Markov process 1 of 14 Course: Theory of Probability II Term: Spring 2015 Instructor: Gordan Zitkovic Lecture 17 Brownian motion as a Markov process Brownian motion is
More informationBoltzmann asymptotics with diffuse reflection boundary conditions.
Boltzmann asymptotics with diffuse reflection boundary conditions. L. Arkeryd and A. Nouri. Key words. Botzmann asymptotics, strong L 1 convergence, diffuse reflection boundary. Mathematics Subject Classification:
More informationSolving PDEs with freefem++
Solving PDEs with freefem++ Tutorials at Basque Center BCA Olivier Pironneau 1 with Frederic Hecht, LJLLUniversity of Paris VI 1 March 13, 2011 Do not forget That everything about freefem++ is at www.freefem.org
More informationIterative Learning Control Analysis and Design I
Iterative Learning Control Analysis and Design I Electronics and Computer Science University of Southampton Southampton, SO17 1BJ, UK etar@ecs.soton.ac.uk http://www.ecs.soton.ac.uk/ Contents Basics Representations
More informationElements of linear algebra
Elements of linear algebra Elements of linear algebra A vector space S is a set (numbers, vectors, functions) which has addition and scalar multiplication defined, so that the linear combination c 1 v
More informationModelling and numerical methods for the diffusion of impurities in a gas
INERNAIONAL JOURNAL FOR NUMERICAL MEHODS IN FLUIDS Int. J. Numer. Meth. Fluids 6; : 6 [Version: /9/8 v.] Modelling and numerical methods for the diffusion of impurities in a gas E. Ferrari, L. Pareschi
More informationMathematical Methods for Neurosciences. ENS  Master MVA Paris 6  Master MathsBio ( )
Mathematical Methods for Neurosciences. ENS  Master MVA Paris 6  Master MathsBio (20142015) Etienne Tanré  Olivier Faugeras INRIA  Team Tosca November 26th, 2014 E. Tanré (INRIA  Team Tosca) Mathematical
More informationThe Gaussian free field, Gibbs measures and NLS on planar domains
The Gaussian free field, Gibbs measures and on planar domains N. Burq, joint with L. Thomann (Nantes) and N. Tzvetkov (Cergy) Université Paris Sud, Laboratoire de Mathématiques d Orsay, CNRS UMR 8628 LAGA,
More informationA quantum heat equation 5th Spring School on Evolution Equations, TU Berlin
A quantum heat equation 5th Spring School on Evolution Equations, TU Berlin Mario Bukal A. Jüngel and D. Matthes ACROSS  Centre for Advanced Cooperative Systems Faculty of Electrical Engineering and Computing
More informationStable Numerical Scheme for the Magnetic Induction Equation with Hall Effect
Stable Numerical Scheme for the Magnetic Induction Equation with Hall Effect Paolo Corti joint work with Siddhartha Mishra ETH Zurich, Seminar for Applied Mathematics 1719th August 2011, Pro*Doc Retreat,
More informationMIT (Spring 2014)
18.311 MIT (Spring 014) Rodolfo R. Rosales May 6, 014. Problem Set # 08. Due: Last day of lectures. IMPORTANT: Turn in the regular and the special problems stapled in two SEPARATE packages. Print your
More informationSolutions Chapter 9. u. (c) u(t) = 1 e t + c 2 e 3 t! c 1 e t 3c 2 e 3 t. (v) (a) u(t) = c 1 e t cos 3t + c 2 e t sin 3t. (b) du
Solutions hapter 9 dode 9 asic Solution Techniques 9 hoose one or more of the following differential equations, and then: (a) Solve the equation directly (b) Write down its phase plane equivalent, and
More informationNonlinear error dynamics for cycled data assimilation methods
Nonlinear error dynamics for cycled data assimilation methods A J F Moodey 1, A S Lawless 1,2, P J van Leeuwen 2, R W E Potthast 1,3 1 Department of Mathematics and Statistics, University of Reading, UK.
More informationarxiv: v2 [math.oc] 30 Sep 2015
Symmetry, Integrability and Geometry: Methods and Applications Moments and Legendre Fourier Series for Measures Supported on Curves Jean B. LASSERRE SIGMA 11 (215), 77, 1 pages arxiv:158.6884v2 [math.oc]
More informationHopping transport in disordered solids
Hopping transport in disordered solids Dominique Spehner Institut Fourier, Grenoble, France Workshop on Quantum Transport, Lexington, March 17, 2006 p. 1 Outline of the talk Hopping transport Models for
More informationFinal: Solutions Math 118A, Fall 2013
Final: Solutions Math 118A, Fall 2013 1. [20 pts] For each of the following PDEs for u(x, y), give their order and say if they are nonlinear or linear. If they are linear, say if they are homogeneous or
More informationDOMAIN OF ATTRACTION: ESTIMATES FOR NONPOLYNOMIAL SYSTEMS VIA LMIS. Graziano Chesi
DOMAIN OF ATTRACTION: ESTIMATES FOR NONPOLYNOMIAL SYSTEMS VIA LMIS Graziano Chesi Dipartimento di Ingegneria dell Informazione Università di Siena Email: chesi@dii.unisi.it Abstract: Estimating the Domain
More informationNonlinear stability of semidiscrete shocks for twosided schemes
Nonlinear stability of semidiscrete shocks for twosided schemes Margaret Beck Boston University Joint work with Hermen Jan Hupkes, Björn Sandstede, and Kevin Zumbrun Setting: semidiscrete conservation
More informationUne décomposition micromacro particulaire pour des équations de type BoltzmannBGK en régime de diffusion
Une décomposition micromacro particulaire pour des équations de type BoltzmannBGK en régime de diffusion Anaïs Crestetto 1, Nicolas Crouseilles 2 et Mohammed Lemou 3 Rennes, 14ème Journée de l équipe
More informationNonlinear Control Systems
Nonlinear Control Systems António Pedro Aguiar pedro@isr.ist.utl.pt 7. Feedback Linearization ISTDEEC PhD Course http://users.isr.ist.utl.pt/%7epedro/ncs1/ 1 1 Feedback Linearization Given a nonlinear
More informationLinear Response and Onsager Reciprocal Relations
Linear Response and Onsager Reciprocal Relations Amir Bar January 1, 013 Based on Kittel, Elementary statistical physics, chapters 3334; Kubo,Toda and Hashitsume, Statistical Physics II, chapter 1; and
More informationAlgebra I Fall 2007
MIT OpenCourseWare http://ocw.mit.edu 18.701 Algebra I Fall 007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.701 007 Geometry of the Special Unitary
More informationStability Analysis for ODEs
Stability Analysis for ODEs Marc R Roussel September 13, 2005 1 Linear stability analysis Equilibria are not always stable Since stable and unstable equilibria play quite different roles in the dynamics
More informationPhase transitions and coarse graining for a system of particles in the continuum
Phase transitions and coarse graining for a system of particles in the continuum Elena Pulvirenti (joint work with E. Presutti and D. Tsagkarogiannis) Leiden University April 7, 2014 Symposium on Statistical
More informationImplementation of Sparse WaveletGalerkin FEM for Stochastic PDEs
Implementation of Sparse WaveletGalerkin FEM for Stochastic PDEs Roman Andreev ETH ZÜRICH / 29 JAN 29 TOC of the Talk Motivation & SetUp Model Problem Stochastic Galerkin FEM Conclusions & Outlook Motivation
More informationSection 12.6: Nonhomogeneous Problems
Section 12.6: Nonhomogeneous Problems 1 Introduction Up to this point all the problems we have considered are we what we call homogeneous problems. This means that for an interval < x < l the problems
More informationReview of last lecture I
Review of last lecture I t f + v x f = Q γ (f,f), (1) where f (x,v,t) is a nonegative function that represents the density of particles in position x at time t with velocity v. In (1) Q γ is the socalled
More information2.3 Damping, phases and all that
2.3. DAMPING, PHASES AND ALL THAT 107 2.3 Damping, phases and all that If we imagine taking our idealized mass on a spring and dunking it in water or, more dramatically, in molasses), then there will be
More information1. INTRODUCTION 2. PROBLEM FORMULATION ROMAI J., 6, 2(2010), 1 13
Contents 1 A product formula approach to an inverse problem governed by nonlinear phasefield transition system. Case 1D Tommaso Benincasa, Costică Moroşanu 1 v ROMAI J., 6, 2(21), 1 13 A PRODUCT FORMULA
More informationInvariances in spectral estimates. ParisEst MarnelaVallée, January 2011
Invariances in spectral estimates Franck Barthe Dario CorderoErausquin ParisEst MarnelaVallée, January 2011 Notation Notation Given a probability measure ν on some Euclidean space, the Poincaré constant
More informationPutzer s Algorithm. Norman Lebovitz. September 8, 2016
Putzer s Algorithm Norman Lebovitz September 8, 2016 1 Putzer s algorithm The differential equation dx = Ax, (1) dt where A is an n n matrix of constants, possesses the fundamental matrix solution exp(at),
More informationPropagating terraces and the dynamics of frontlike solutions of reactiondiffusion equations on R
Propagating terraces and the dynamics of frontlike solutions of reactiondiffusion equations on R P. Poláčik School of Mathematics, University of Minnesota Minneapolis, MN 55455 Abstract We consider semilinear
More information1. Plurisubharmonic functions and currents The first part of the homework studies some properties of PSH functions.
MATH 263: PROBLEM SET 2: PSH FUNCTIONS, HORMANDER S ESTIMATES AND VANISHING THEOREMS 1. Plurisubharmonic functions and currents The first part of the homework studies some properties of PSH functions.
More informationA LévyFokkerPlanck equation: entropies and convergence to equilibrium
1/ 22 A LévyFokkerPlanck equation: entropies and convergence to equilibrium I. Gentil CEREMADE, Université ParisDauphine International Conference on stochastic Analysis and Applications Hammamet, Tunisia,
More informationElements of Positive Definite Kernel and Reproducing Kernel Hilbert Space
Elements of Positive Definite Kernel and Reproducing Kernel Hilbert Space Statistical Inference with Reproducing Kernel Hilbert Space Kenji Fukumizu Institute of Statistical Mathematics, ROIS Department
More informationQuantum ergodicity. Nalini Anantharaman. 22 août Université de Strasbourg
Quantum ergodicity Nalini Anantharaman Université de Strasbourg 22 août 2016 I. Quantum ergodicity on manifolds. II. QE on (discrete) graphs : regular graphs. III. QE on graphs : other models, perspectives.
More informationTHREEPOINT BOUNDARY VALUE PROBLEMS WITH SOLUTIONS THAT CHANGE SIGN
JOURNAL OF INTEGRAL EQUATIONS AND APPLICATIONS Volume 15, Number 1, Spring 23 THREEPOINT BOUNDARY VALUE PROBLEMS WITH SOLUTIONS THAT CHANGE SIGN G. INFANTE AND J.R.L. WEBB ABSTRACT. Using the theory of
More informationDynamics of the di usive Nicholson s blow ies equation with distributed delay
Proceedings of the Royal Society of Edinburgh, 3A, 275{29, 2 Dynamics of the di usive Nicholson s blow ies equation with distributed delay S. A. Gourley Department of Mathematics and Statistics, University
More information4 Film Extension of the Dynamics: Slowness as Stability
4 Film Extension of the Dynamics: Slowness as Stability 4.1 Equation for the Film Motion One of the difficulties in the problem of reducing the description is caused by the fact that there exists no commonly
More informationMath 127: Course Summary
Math 27: Course Summary Rich Schwartz October 27, 2009 General Information: M27 is a course in functional analysis. Functional analysis deals with normed, infinite dimensional vector spaces. Usually, these
More informationDefinition and basic properties of heat kernels I, An introduction
Definition and basic properties of heat kernels I, An introduction Zhiqin Lu, Department of Mathematics, UC Irvine, Irvine CA 92697 April 23, 2010 In this lecture, we will answer the following questions:
More informationIntegration of Vlasovtype equations
Alexander Ostermann University of Innsbruck, Austria Joint work with Lukas Einkemmer Verona, April/May 2017 Plasma the fourth state of matter 99% of the visible matter in the universe is made of plasma
More informationThis is a Gaussian probability centered around m = 0 (the most probable and mean position is the origin) and the mean square displacement m 2 = n,or
Physics 7b: Statistical Mechanics Brownian Motion Brownian motion is the motion of a particle due to the buffeting by the molecules in a gas or liquid. The particle must be small enough that the effects
More informationz + az + bz + 12 = 0, where a and b are real numbers, is
016 IJC H Math 9758 Promotional Examination 1 One root of the equation z = 1 i z + az + bz + 1 = 0, where a and b are real numbers, is. Without using a calculator, find the values of a and b and the other
More informationReproducing formulas associated with symbols
Reproducing formulas associated with symbols Filippo De Mari Ernesto De Vito Università di Genova, Italy Modern Methods of TimeFrequency Analysis II Workshop on Applied Coorbit space theory September
More informationMath 127C, Spring 2006 Final Exam Solutions. x 2 ), g(y 1, y 2 ) = ( y 1 y 2, y1 2 + y2) 2. (g f) (0) = g (f(0))f (0).
Math 27C, Spring 26 Final Exam Solutions. Define f : R 2 R 2 and g : R 2 R 2 by f(x, x 2 (sin x 2 x, e x x 2, g(y, y 2 ( y y 2, y 2 + y2 2. Use the chain rule to compute the matrix of (g f (,. By the chain
More informationThe parabolic Anderson model on Z d with timedependent potential: Frank s works
Weierstrass Institute for Applied Analysis and Stochastics The parabolic Anderson model on Z d with timedependent potential: Frank s works Based on Frank s works 2006 2016 jointly with Dirk Erhard (Warwick),
More information