Homogenization of reaction-diffusion processes in a two-component porous medium with a non-linear flux-condition on the interface
|
|
- Gabriel Taylor
- 5 years ago
- Views:
Transcription
1 Homogenzaton of reacton-dffuson processes n a two-component porous medum wth a non-lnear flux-condton on the nterface Internatonal Conference on Numercal and Mathematcal Modelng of Flow and Transport n Porous Meda, 09/29/2014 Jont work wth P. Knabner and M. Neuss-Radu Markus Gahn Unversty Erlangen-Nuremberg
2 Motvaton of the Model Sketch of the carbohydrate-metabolsm n a plant cell: Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 2
3 Mcroscopc doman - A porous medum wth two perodcally dstrbuted components separated by an nterface s consdered - Metaboltes can be transported through the nterface The flux over the nterface s contnuous and gven by a nonlnear mult-speces reacton rate Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 3
4 Notatons - Y = (0, 1) n, Ω = [a, b] wth a, b Z n and a < b - Y 2 Y open Lpschtz-doman and Y 1 = Y \ Y 2, Γ = Y 2 and Y 1 s connected - Ω ɛ j = { x Ω R n x ɛy k j for k Z n} for j = 1, 2 and ɛ > 0 wth ɛ 1 N Ω ɛ 1 s connected Ω ɛ 2 s dsconnected - Γ ɛ = { x Ω x ɛγ k for k Z n} - u j,ɛ (j = 1, 2, = 1,... m) denotes the concentraton of the -th speces n the doman Ω ɛ j u 1,ɛ and u 2,ɛ descrbe the same speces n the dfferent domans Ω ɛ 1 and Ωɛ 2 Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 4
5 Mcroscopc equatons Fnd u ɛ = (u 1,ɛ, u 2,ɛ ) = (u 1,ɛ 1,..., u1,ɛ m, u 2,ɛ 1,..., u2,ɛ m ), wth u 1,ɛ : (0, T ) Ω ɛ 1 R m and u 2,ɛ : (0, T ) Ω ɛ 2 R m, such that D 1 u1,ɛ t u j,ɛ D j uj,ɛ = f j,ɛ (t, x, u j,ɛ ) n (0, T ) Ω ɛ j, ν 2 = D 2 u2,ɛ ν 2 =ɛh (u 1,ɛ, u 2,ɛ ) on (0, T ) Γ ɛ, D 1 u 1,ɛ ν 1 = 0 on (0, T ) Ω, u j,ɛ (0) = u j,0 n L 2 (Ω ɛ j ), where ν j denotes the outer unt normal on Ω ɛ j. Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 5
6 Mult-Substrate Reactons - Mult-substrate reactons catalyzed by enzymes are of the form X X m1 Y Y m2 - Reacton-rate under the quas-steady state assumpton v(x, Y ) = a m 1 k=1 X k b m 2 k=1 Y k p(x, Y ) wth p(x, Y ) = α m 1, β m 2 c α,β X α Y β and c α,β 0 Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 6
7 Assumptons on the Data - f j,ɛ and h are Lpschtz-contnuous - There exsts M, A > 0, such that for all z, w, v M holds f j,ɛ (t, x, z) Az, h (v, w) Av, h (v, w) Aw - For (t, x, z) (0, T ) Ω R n holds m f j,ɛ (t, x, z)(z ) C =1 - For (v, w) R n R n holds h (v, w) [(v ) (w ) ] C - The ntal values satsfy 0 u j,0 M m (z ) 2 =1 m ( (v ) 2 + (w ) 2) =1 Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 7
8 Mcroscopc Equatons - Varatonal Formulaton Fnd wth (u 1,ɛ, u 2,ɛ ) L 2 ((0, T ), H 1 (Ω ɛ 1, R m )) L 2 ((0, T ), H 1 (Ω ɛ 2, R m )) ( t u 1,ɛ, t u 2,ɛ ) L 2 ((0, T ), H 1 (Ω ɛ 1, R m ) ) L 2 ((0, T ), H 1 (Ω ɛ 2, R m ) ), such that for all φ j H 1 (Ω ɛ j ) and a.e. t (0, T ) holds t u j,ɛ, φ j Ω ɛ j + D j ( uj,ɛ, φ j ) Ω ɛ j = (f j,ɛ (u j,ɛ ), φ j ) Ω ɛ j ( 1) j ɛ(h(u 1,ɛ, u 2,ɛ ), φ j ) Γɛ, wth - (, ) Ω ɛ j s the nner-product n L 2 (Ω ɛ j ) or L 2 (Ω ɛ j ) n, respectvely -, Ω ɛ j s the dualty parng on H 1 (Ω ɛ j ) H 1 (Ω ɛ j ). Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 8
9 Exstence and Unqueness Theorem - There exsts a unque weak soluton u ɛ = (u 1,ɛ, u 2,ɛ ) wth u j,ɛ L 2 ((0, T ), H 1 (Ω ɛ j )) m and t u j,ɛ L 2 ((0, T ), H 1 (Ω ɛ j ) ) m - The soluton s non-negatve - The followng a-pror estmates are fulflled u j,ɛ L ((0,T ) Ω ɛ j ) + u j,ɛ L2 ((0,T ),H 1 (Ω ɛ j )) + t u j,ɛ L2 ((0,T ),H 1 (Ω ɛ j ) ) + ɛ u j,ɛ L2 ((0,T ),L 2 (Γ ɛ )) C Proof: Solve the Problem for a lnear flux-condton on the boundary and use Schaefer s fxed pont theorem Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 9
10 Two-scale Convergence ([92Allare] 1, [89Nguetseng] 2 ) - A sequence u ɛ L 2 ((0, T ) Ω) converges n the two-scale sense to u 0 L 2 ((0, T ) Ω Y ), f for every φ C ( ( )) [0, T ] Ω, C per Y holds T ( lm u ɛ (t, x)φ t, x, x ) T dxdt = u 0 (t, x, y)φ(t, x, y)dydxdt ɛ 0 ɛ 0 Ω - A two-scale convergent sequence u ɛ convergences strongly n the two-scale sense to u 0, f addtonally lm ɛ 0 u ɛ L2 ((0,T ) Ω) = u 0 L2 ((0,T ) Ω Y ) 0 Ω Y 1 G. Allare, Homogenzaton and two-scale convergence, SIAM J. Math. Anal.,23 (1992), pp G. Nguetseng, A general convergence result for a functonal related to the theory of homogenzaton, SIAM J. Math. Anal., 20 (1989), pp Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 10
11 Two-scale Convergence on Γ ɛ ([96Allare et al.] 3, [96Neuss-Radu] 4 ) - A sequence u ɛ L 2 ((0, T ) Γ ɛ ) converges n the two-scale sense on the surface Γ ɛ to a lmt u 0 L 2 ((0, T ) Ω Γ), f for every φ C ( [0, T ] Ω, C per (Γ) ) holds T lm ɛ ɛ 0 0 Γ ɛ u ɛ (t, x)φ ( t, x, x ɛ ) dσdt = T 0 Ω Γ u 0 (t, x, y)φ(t, x, y)dσ y dxdt - A two-scale convergent sequence u ɛ on Γ ɛ converges strongly n the two-scale sense on Γ ɛ, f addtonally holds ɛ uɛ L2 ((0,T ) Γ ɛ ) = u 0 L2 ((0,T ) Ω Γ) lm ɛ 0 3 G. Allare, A. Damlaman, U. Hornung, Two-scale convergence on perodc surfaces and applcatons, Proceedngs of the nternatonal conference on mathematcal modellng of flow through porous meda, A. Bourgeat et al., eds., World Scentfc, Sngapore (1996), pp M. Neuss-Radu, Some Extensons of Two-Scale Convergence, C. R. Acad. Sc. Pars Sér. I Math., 322 (1996), pp Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 11
12 The Unfoldng Operator ([02,06,08Coranescu et al.] 567 ) Consder the unfoldng operator L ɛ on three dfferent domans: L ɛ : L 2 ((0, T ) Ω) L 2 ((0, T ) Ω Y ) L ɛ : L 2 ((0, T ) Ω ɛ j ) L 2 ((0, T ) Ω Y j ) L ɛ : L 2 ((0, T ) Γ ɛ ) L 2 ((0, T ) Ω Γ) defned by ([ x ] L ɛ u ɛ (t, x, y) := u ɛ (t, ɛ ɛ )) + y 5 D. Coranescu, A. Damlaman, G. Grso, Perodc unfoldng and homogenzaton, C. R. Acad. Sc. Pars Sér. 1, 335 (2002), pp D. Coranescu, P. Donato, R. Zak, The perodc unfoldng method n perforated domans, Port. Math. (N.S.), 63 (2006), pp D.Coranescu, A. Damlaman, G. Grso, The perodc unfoldng method n homogenzaton, SIAM J. Math. Anal., 40 (2008), pp Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 12
13 Unfoldng and Two-Scale-Convergence ([96Bourgeat et al.] 8 ) Lemma () Let {u ɛ } ɛ>0 L 2 ((0, T ) Ω) be bounded. Then are equvalent a) u ɛ u 0 weakly/strongly n the two-scale sense b) L ɛ u ɛ u 0 weakly/strongly n L 2 ((0, T ) Ω Y ) () Let {u ɛ } ɛ>0 L 2 ((0, T ) Γ ɛ ) wth ɛ u ɛ L2 ((0,T ) Γ ɛ ) C. Then are equvalent a) u ɛ u 0 weakly/strongly n the two-scale sense on Γ ɛ b) L ɛ u ɛ u 0 weakly/strongly n L 2 ((0, T ) Ω Γ) 8 A. Bourgeat, S. Luckhaus, A. Mkelć, Convergence of the homogenzaton process for a double-porosty model of mmscble two-phase flow, SIAM J. Math. Anal., 27 (1996), pp Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 13
14 Extensons to the doman Ω - It exsts F ɛ : L 2 ((0, T ), H 1 (Ω ɛ 1)) L 2 ((0, T ), H 1 (Ω)) wth F ɛ u ɛ L2 ((0,T ),H 1 (Ω)) C u ɛ L2 ((0,T ),H 1 (Ω ɛ 1 )) Wrte ũ ɛ = F ɛ u ɛ - For u ɛ L 2 ((0, T ) Ω ɛ 2) denote by u ɛ the zero extenson to Ω - For u ɛ L 2 ((0, T ), H 1 (Ω ɛ j )) wth t u ɛ L 2 ((0, T ), H 1 (Ω ɛ j ) ) defne t u ɛ := t ( χ Ω ɛ j u ɛ ) L 2 ((0, T ), H 1 (Ω) ) wth the characterstc functon χ Ω ɛ j on Ω ɛ j Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 14
15 Convergence results for u 1,ɛ Theorem There exsts a subsequence of u 1,ɛ and functons u 1,0 L 2 ((0, T ), H 1 (Ω)) wth t u 1,0 L 2 ((0, T ), H 1 (Ω) ) and u 1,1 L ( 2 (0, T ) Ω, Hper(Y 1 )/R ) for = 1,..., m, such that ũ 1,ɛ u 1,0 strongly n L 2 ((0, T ), L 2 (Ω)) ũ 1,ɛ x u 1,0 + y u 1,1 n the two-scale sense u 1,ɛ Γɛ u 1,0 strongly n the two-scale sense on Γ ɛ t u 1,ɛ Y 1 t u 1,0 weakly n L 2 ((0, T ), H 1 (Ω) ) Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 15
16 Convergence results for u 2,ɛ Theorem There exsts a functon u 2,0 L 2 ((0, T ) Ω) wth t u 2,0 L 2 ((0, T ), H 1 (Ω) ), such that up to a subsequence u 2,ɛ χ Y2 u 2,0 strongly n the two-scale sense u 2,ɛ 0 n the two-scale sense u 2,ɛ Γɛ u 2,0 strongly n the two-scale sense on Γ ɛ t u 2,ɛ Y 2 t u 2,0 weakly n L 2 ((0, T ), H 1 (Ω) ) Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 16
17 Convergence results for u 2,ɛ Theorem There exsts a functon u 2,0 L 2 ((0, T ) Ω) wth t u 2,0 L 2 ((0, T ), H 1 (Ω) ), such that up to a subsequence u 2,ɛ χ Y2 u 2,0 strongly n the two-scale sense u 2,ɛ 0 n the two-scale sense u 2,ɛ Γɛ u 2,0 strongly n the two-scale sense on Γ ɛ t u 2,ɛ Y 2 t u 2,0 weakly n L 2 ((0, T ), H 1 (Ω) ) Problems: - No good extenson operator from L 2 ((0, T ), H 1 (Ω ɛ 2)) nto L 2 ((0, T ), H 1 (Ω)) Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 16
18 Convergence results for u 2,ɛ Theorem There exsts a functon u 2,0 L 2 ((0, T ) Ω) wth t u 2,0 L 2 ((0, T ), H 1 (Ω) ), such that up to a subsequence u 2,ɛ χ Y2 u 2,0 strongly n the two-scale sense u 2,ɛ 0 n the two-scale sense u 2,ɛ Γɛ u 2,0 strongly n the two-scale sense on Γ ɛ t u 2,ɛ Y 2 t u 2,0 weakly n L 2 ((0, T ), H 1 (Ω) ) Problems: - No good extenson operator from L 2 ((0, T ), H 1 (Ω ɛ 2)) nto L 2 ((0, T ), H 1 (Ω)) - The unfolded equaton has an ɛ-scalng n the dffuson term Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 16
19 Convergence results for u 2,ɛ Theorem There exsts a functon u 2,0 L 2 ((0, T ) Ω) wth t u 2,0 L 2 ((0, T ), H 1 (Ω) ), such that up to a subsequence u 2,ɛ χ Y2 u 2,0 strongly n the two-scale sense u 2,ɛ 0 n the two-scale sense u 2,ɛ Γɛ u 2,0 strongly n the two-scale sense on Γ ɛ t u 2,ɛ Y 2 t u 2,0 weakly n L 2 ((0, T ), H 1 (Ω) ) Problems: - No good extenson operator from L 2 ((0, T ), H 1 (Ω ɛ 2)) nto L 2 ((0, T ), H 1 (Ω)) - The unfolded equaton has an ɛ-scalng n the dffuson term - We can t use shfts on the boundary Γ ɛ resp. Γ (.e. a standard Kolmogorov-argument s not possble) Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 16
20 a) Proof: u 2,ɛ 0 n the two-scale sense - Standard compactness results mply the exstence of ξ L 2 ((0, T ) Ω Y ) n, such that (subsequence) u 2,ɛ χ Y2 ξ weakly n the two-scale sense Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 17
21 a) Proof: u 2,ɛ 0 n the two-scale sense - Standard compactness results mply the exstence of ξ L 2 ((0, T ) Ω Y ) n, such that (subsequence) u 2,ɛ χ Y2 ξ weakly n the two-scale sense - ξ can be represented by a gradent n Y 2 : wth p L 2 ((0, T ) Ω, H 1 (Y 2 )/R). ξ = y p n L 2 ((0, T ) Ω, L 2 (Y 2 )) Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 17
22 a) Proof: u 2,ɛ 0 n the two-scale sense - Standard compactness results mply the exstence of ξ L 2 ((0, T ) Ω Y ) n, such that (subsequence) u 2,ɛ χ Y2 ξ weakly n the two-scale sense - ξ can be represented by a gradent n Y 2 : wth p L 2 ((0, T ) Ω, H 1 (Y 2 )/R). - It holds ξ = y p n L 2 ((0, T ) Ω, L 2 (Y 2 )) for all φ H 1 (Y 2 ) ξ = 0 n Y 2 Y 2 y p y φdy = 0 Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 17
23 b) Proof: u 2,ɛ Γɛ u 2,0 strongly n the two-scale sense on Γ ɛ - The functon δu of u for l Z n s gven by - For U R n bounded and h > 0 set δu(t, x) = u(t, x + lɛ) u(t, x) U h = {x U : dst(x, U) > h} Lemma For almost every t (0, T ) and all l Z n wth lɛ < h and 0 < h < 1 2 t holds that δu 2,ɛ (t) 2 L 2 (Ω ɛ 2,h ) C (ɛ 2 + m k=1 ( δu 2 k,0 2 L 2 (Ω ɛ 2,h ) + δu1,ɛ k 2 L 2 ((0,T ),L 2 (Ω ɛ 1,h )) ) ) Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 18
24 b) Proof: u 2,ɛ Γɛ u 2,0 strongly n the two-scale sense on Γ ɛ - For L ɛ u 2,ɛ L 2 ((0, T ) Ω, H 1/2 (Γ)) holds wth ξ R n Lɛ u 2,ɛ (, + ξ) L ɛ u 2,ɛ (, ) L2 ((0,T ) Ω,L 2 (Γ)) C h 2 + ɛ 2 + ( ( [ ])) ξ u2,ɛ, + ɛ m + u 2,ɛ (, ) ɛ m {0,1} n 2 L 2 ((0,T ) Ω ɛ 2,h Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 19
25 b) Proof: u 2,ɛ Γɛ u 2,0 strongly n the two-scale sense on Γ ɛ - For L ɛ u 2,ɛ L 2 ((0, T ) Ω, H 1/2 (Γ)) holds wth ξ R n Lɛ u 2,ɛ (, + ξ) L ɛ u 2,ɛ (, ) L2 ((0,T ) Ω,L 2 (Γ)) C h 2 + ɛ 2 + ( ( [ ])) ξ u2,ɛ, + ɛ m + u 2,ɛ (, ) ɛ m {0,1} n 2 L 2 ((0,T ) Ω ɛ 2,h - Ths mples for δ > 0, δ = (δ,..., δ) R n L ɛ u 2,ɛ ( + δ, + δ) L ɛ u 2,ɛ (, ) L2 (W δ,l 2 (Γ)) δ 0 0 unformly n ɛ wth W δ = (0, T δ) (Ω (Ω δ)) Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 19
26 b) Proof: u 2,ɛ Γɛ u 2,0 strongly n the two-scale sense on Γ ɛ - For L ɛ u 2,ɛ L 2 ((0, T ) Ω, H 1/2 (Γ)) holds wth ξ R n Lɛ u 2,ɛ (, + ξ) L ɛ u 2,ɛ (, ) L2 ((0,T ) Ω,L 2 (Γ)) C h 2 + ɛ 2 + ( ( [ ])) ξ u2,ɛ, + ɛ m + u 2,ɛ (, ) ɛ m {0,1} n 2 L 2 ((0,T ) Ω ɛ 2,h - Ths mples for δ > 0, δ = (δ,..., δ) R n L ɛ u 2,ɛ ( + δ, + δ) L ɛ u 2,ɛ (, ) L2 (W δ,l 2 (Γ)) δ 0 0 unformly n ɛ wth W δ = (0, T δ) (Ω (Ω δ)) - For every measurable subset A (0, T ) Ω the set { } L ɛ u 2,ɛ (t, x, )dxdt H 1/2 (Γ) s relatvely compact n L 2 (Γ) A ɛ>0 Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 19
27 b) Proof: u 2,ɛ Γɛ u 2,0 strongly n the two-scale sense on Γ ɛ Now conclude wth the followng Theorem Theorem Let F L 2 (W, B) wth a Banach-space B and a W R n a cubod. Then F s relatvely compact n L 2 (W, B) f and only f for δ > 0 and δ = (δ,..., δ) R n holds () For every measurable set A W the set { A udx u F } s relatvely compact n B. () u( + δ 0 δ) u( ) 0 unformly n F. L2 (W (W δ),b) Proof: Same arguments for the 1-dmensonal case as n [Smon] 9. = {L ɛ u 2,ɛ } ɛ>0 s relatvely compact n L 2 ((0, T ) Ω, L 2 (Γ)) 9 J. Smon, Compact Sets n the Space L p (0, T ; B), Ann. Mat. Pura Appl., 146 (1987), pp Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 20
28 The Macroscopc Equatons Theorem The functon u 1,0 Y 1 t u 1,0 and for u 2,0 s a weak soluton of ( ) D u 1,0 ( = Y 1 f ) 1 u 1,0 + Γ h ( u 1,0, u 2,0) n (0, T ) Ω D u 1,0 ν = 0 on (0, T ) Ω u 1,0 (0) = u 1,0 n L 2 (Ω) holds the ordnary dfferental equaton ( ) u 2,0 Γ h ( u 1,0, u 2,0) n (0, T ) Ω Y 2 t u 2,0 = Y 2 f 2 u 2,0 = u 2,0 n L 2 (Ω) The dffuson-term vanshes n the upscaled equatons Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 21
29 The Macroscopc Equatons Theorem The homogenzed matrx D R n n s gven by (D ) k,l = D 1 ( w k + e k ) ( w l + e l ) dy, Y 1 where the w k, k = 1,..., n, are the unque solutons of the cell problem w = 0 n Y 1 w ν = ν on Γ w s Y -perodc and w dy = 0 Y 1 Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 22
30 The Macroscopc Equatons Theorem The homogenzed matrx D R n n s gven by (D ) k,l = D 1 ( w k + e k ) ( w l + e l ) dy, Y 1 where the w k, k = 1,..., n, are the unque solutons of the cell problem w s Y -perodc and Theorem The soluton (u 1,0, u 2,0 ) s unque. w = 0 n Y 1 w ν = ν on Γ Y 1 w dy = 0 Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 22
31 Concluson and Outlook We derved a homogenzed model for a two component medum for whch one component s dsconnected We consdered nonlnear transmsson condtons at the nterface between the two meda The rgorous dervaton of the macroscopc model nvolved the strong two-scale convergence for functons defned on perodc surfaces. To obtan ths, we used a compactness crteron of Smon, whch we generalzed to functons defned on a rectancle n R n wth values n a Banach-space The presented model descrbes only a part of the carbohydrate metabolsm n the plant cell. In an upcomng paper the full carbohydrate metabolsm s modelled and analysed. Markus Gahn Unversty Erlangen-Nuremberg Homogenzaton of a non-lnear transport process 23
32 Thank you for your attenton!
ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EQUATION
Advanced Mathematcal Models & Applcatons Vol.3, No.3, 2018, pp.215-222 ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EUATION
More informationConvexity preserving interpolation by splines of arbitrary degree
Computer Scence Journal of Moldova, vol.18, no.1(52), 2010 Convexty preservng nterpolaton by splnes of arbtrary degree Igor Verlan Abstract In the present paper an algorthm of C 2 nterpolaton of dscrete
More informationUniqueness of Weak Solutions to the 3D Ginzburg- Landau Model for Superconductivity
Int. Journal of Math. Analyss, Vol. 6, 212, no. 22, 195-114 Unqueness of Weak Solutons to the 3D Gnzburg- Landau Model for Superconductvty Jshan Fan Department of Appled Mathematcs Nanjng Forestry Unversty
More informationGeneral viscosity iterative method for a sequence of quasi-nonexpansive mappings
Avalable onlne at www.tjnsa.com J. Nonlnear Sc. Appl. 9 (2016), 5672 5682 Research Artcle General vscosty teratve method for a sequence of quas-nonexpansve mappngs Cuje Zhang, Ynan Wang College of Scence,
More informationAsymptotics of the Solution of a Boundary Value. Problem for One-Characteristic Differential. Equation Degenerating into a Parabolic Equation
Nonl. Analyss and Dfferental Equatons, ol., 4, no., 5 - HIKARI Ltd, www.m-har.com http://dx.do.org/.988/nade.4.456 Asymptotcs of the Soluton of a Boundary alue Problem for One-Characterstc Dfferental Equaton
More informationOn a nonlinear compactness lemma in L p (0, T ; B).
On a nonlnear compactness lemma n L p (, T ; B). Emmanuel Matre Laboratore de Matématques et Applcatons Unversté de Haute-Alsace 4, rue des Frères Lumère 6893 Mulouse E.Matre@ua.fr 3t February 22 Abstract
More informationModelli Clamfim Equazioni differenziali 7 ottobre 2013
CLAMFIM Bologna Modell 1 @ Clamfm Equazon dfferenzal 7 ottobre 2013 professor Danele Rtell danele.rtell@unbo.t 1/18? Ordnary Dfferental Equatons A dfferental equaton s an equaton that defnes a relatonshp
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationNumerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
More information2.3 Nilpotent endomorphisms
s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms
More informationThe Two-scale Finite Element Errors Analysis for One Class of Thermoelastic Problem in Periodic Composites
7 Asa-Pacfc Engneerng Technology Conference (APETC 7) ISBN: 978--6595-443- The Two-scale Fnte Element Errors Analyss for One Class of Thermoelastc Problem n Perodc Compostes Xaoun Deng Mngxang Deng ABSTRACT
More informationA CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS
Journal of Mathematcal Scences: Advances and Applcatons Volume 25, 2014, Pages 1-12 A CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS JIA JI, WEN ZHANG and XIAOFEI QI Department of Mathematcs
More informationLinear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
More informationFinite Element Modelling of truss/cable structures
Pet Schreurs Endhoven Unversty of echnology Department of Mechancal Engneerng Materals echnology November 3, 214 Fnte Element Modellng of truss/cable structures 1 Fnte Element Analyss of prestressed structures
More informationGeorgia Tech PHYS 6124 Mathematical Methods of Physics I
Georga Tech PHYS 624 Mathematcal Methods of Physcs I Instructor: Predrag Cvtanovć Fall semester 202 Homework Set #7 due October 30 202 == show all your work for maxmum credt == put labels ttle legends
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationMMA and GCMMA two methods for nonlinear optimization
MMA and GCMMA two methods for nonlnear optmzaton Krster Svanberg Optmzaton and Systems Theory, KTH, Stockholm, Sweden. krlle@math.kth.se Ths note descrbes the algorthms used n the author s 2007 mplementatons
More informationBOUNDEDNESS OF THE RIESZ TRANSFORM WITH MATRIX A 2 WEIGHTS
BOUNDEDNESS OF THE IESZ TANSFOM WITH MATIX A WEIGHTS Introducton Let L = L ( n, be the functon space wth norm (ˆ f L = f(x C dx d < For a d d matrx valued functon W : wth W (x postve sem-defnte for all
More informationA new Approach for Solving Linear Ordinary Differential Equations
, ISSN 974-57X (Onlne), ISSN 974-5718 (Prnt), Vol. ; Issue No. 1; Year 14, Copyrght 13-14 by CESER PUBLICATIONS A new Approach for Solvng Lnear Ordnary Dfferental Equatons Fawz Abdelwahd Department of
More informationDeriving the X-Z Identity from Auxiliary Space Method
Dervng the X-Z Identty from Auxlary Space Method Long Chen Department of Mathematcs, Unversty of Calforna at Irvne, Irvne, CA 92697 chenlong@math.uc.edu 1 Iteratve Methods In ths paper we dscuss teratve
More informationThe Finite Element Method: A Short Introduction
Te Fnte Element Metod: A Sort ntroducton Wat s FEM? Te Fnte Element Metod (FEM) ntroduced by engneers n late 50 s and 60 s s a numercal tecnque for solvng problems wc are descrbed by Ordnary Dfferental
More informationGroup Analysis of Ordinary Differential Equations of the Order n>2
Symmetry n Nonlnear Mathematcal Physcs 997, V., 64 7. Group Analyss of Ordnary Dfferental Equatons of the Order n> L.M. BERKOVICH and S.Y. POPOV Samara State Unversty, 4430, Samara, Russa E-mal: berk@nfo.ssu.samara.ru
More informationSELECTED SOLUTIONS, SECTION (Weak duality) Prove that the primal and dual values p and d defined by equations (4.3.2) and (4.3.3) satisfy p d.
SELECTED SOLUTIONS, SECTION 4.3 1. Weak dualty Prove that the prmal and dual values p and d defned by equatons 4.3. and 4.3.3 satsfy p d. We consder an optmzaton problem of the form The Lagrangan for ths
More informationAppendix B. The Finite Difference Scheme
140 APPENDIXES Appendx B. The Fnte Dfference Scheme In ths appendx we present numercal technques whch are used to approxmate solutons of system 3.1 3.3. A comprehensve treatment of theoretcal and mplementaton
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationOn a direct solver for linear least squares problems
ISSN 2066-6594 Ann. Acad. Rom. Sc. Ser. Math. Appl. Vol. 8, No. 2/2016 On a drect solver for lnear least squares problems Constantn Popa Abstract The Null Space (NS) algorthm s a drect solver for lnear
More informationMechanics Physics 151
Mechancs Physcs 151 Lecture 3 Lagrange s Equatons (Goldsten Chapter 1) Hamlton s Prncple (Chapter 2) What We Dd Last Tme! Dscussed mult-partcle systems! Internal and external forces! Laws of acton and
More informationLINEAR INTEGRAL EQUATIONS OF VOLTERRA CONCERNING HENSTOCK INTEGRALS
Real Analyss Exchange Vol. (),, pp. 389 418 M. Federson and R. Bancon, Insttute of Mathematcs and Statstcs, Unversty of São Paulo, CP 66281, 05315-970. e-mal: federson@cmc.sc.usp.br and bancon@me.usp.br
More informationModule 1 : The equation of continuity. Lecture 1: Equation of Continuity
1 Module 1 : The equaton of contnuty Lecture 1: Equaton of Contnuty 2 Advanced Heat and Mass Transfer: Modules 1. THE EQUATION OF CONTINUITY : Lectures 1-6 () () () (v) (v) Overall Mass Balance Momentum
More information8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 493 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces you have studed thus far n the text are real vector spaces because the scalars
More informationON THE EXTENDED HAAGERUP TENSOR PRODUCT IN OPERATOR SPACES. 1. Introduction
ON THE EXTENDED HAAGERUP TENSOR PRODUCT IN OPERATOR SPACES TAKASHI ITOH AND MASARU NAGISA Abstract We descrbe the Haagerup tensor product l h l and the extended Haagerup tensor product l eh l n terms of
More informationFirst day August 1, Problems and Solutions
FOURTH INTERNATIONAL COMPETITION FOR UNIVERSITY STUDENTS IN MATHEMATICS July 30 August 4, 997, Plovdv, BULGARIA Frst day August, 997 Problems and Solutons Problem. Let {ε n } n= be a sequence of postve
More informationClassification as a Regression Problem
Target varable y C C, C,, ; Classfcaton as a Regresson Problem { }, 3 L C K To treat classfcaton as a regresson problem we should transform the target y nto numercal values; The choce of numercal class
More informationAffine and Riemannian Connections
Affne and Remannan Connectons Semnar Remannan Geometry Summer Term 2015 Prof Dr Anna Wenhard and Dr Gye-Seon Lee Jakob Ullmann Notaton: X(M) space of smooth vector felds on M D(M) space of smooth functons
More informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More informationSome modelling aspects for the Matlab implementation of MMA
Some modellng aspects for the Matlab mplementaton of MMA Krster Svanberg krlle@math.kth.se Optmzaton and Systems Theory Department of Mathematcs KTH, SE 10044 Stockholm September 2004 1. Consdered optmzaton
More informationThe Analytical Solution of a System of Nonlinear Differential Equations
Int. Journal of Math. Analyss, Vol. 1, 007, no. 10, 451-46 The Analytcal Soluton of a System of Nonlnear Dfferental Equatons Yunhu L a, Fazhan Geng b and Mnggen Cu b1 a Dept. of Math., Harbn Unversty Harbn,
More information2nd International Conference on Electronics, Network and Computer Engineering (ICENCE 2016)
nd Internatonal Conference on Electroncs, Network and Computer Engneerng (ICENCE 6) Postve solutons of the fourth-order boundary value problem wth dependence on the frst order dervatve YuanJan Ln, a, Fe
More informationExistence results for a fourth order multipoint boundary value problem at resonance
Avalable onlne at www.scencedrect.com ScenceDrect Journal of the Ngeran Mathematcal Socety xx (xxxx) xxx xxx www.elsever.com/locate/jnnms Exstence results for a fourth order multpont boundary value problem
More informationSalmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2
Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to
More informationA Hybrid Variational Iteration Method for Blasius Equation
Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 10, Issue 1 (June 2015), pp. 223-229 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) A Hybrd Varatonal Iteraton Method
More informationCHAPTER II THEORETICAL BACKGROUND
3 CHAPTER II THEORETICAL BACKGROUND.1. Lght Propagaton nsde the Photonc Crystal The frst person that studes the one dmenson photonc crystal s Lord Raylegh n 1887. He showed that the lght propagaton depend
More informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More informationAppendix for Causal Interaction in Factorial Experiments: Application to Conjoint Analysis
A Appendx for Causal Interacton n Factoral Experments: Applcaton to Conjont Analyss Mathematcal Appendx: Proofs of Theorems A. Lemmas Below, we descrbe all the lemmas, whch are used to prove the man theorems
More informationProblem adapted reduced models based on Reaction-Diffusion Manifolds (REDIMs)
Problem adapted reduced models based on Reacton-Dffuson Manfolds (REDIMs) V Bykov, U Maas Thrty-Second Internatonal Symposum on ombuston, Montreal, anada, 3-8 August, 8 Problem Statement: Smulaton of reactng
More informationOn the Gradgrad and divdiv Complexes, and a Related Decomposition Result for Biharmonic Problems in 3D: Part 2
RICAM Specal Semester, Workshop 1: Analyss and Numercs of Acoustc and Electromagnetc Problems Lnz, Austra, October 17, 216 On the Gradgrad and dvdv Complexes, and a Related Decomposton Result for Bharmonc
More informationLecture Notes on Linear Regression
Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume
More informationMATH 5630: Discrete Time-Space Model Hung Phan, UMass Lowell March 1, 2018
MATH 5630: Dscrete Tme-Space Model Hung Phan, UMass Lowell March, 08 Newton s Law of Coolng Consder the coolng of a well strred coffee so that the temperature does not depend on space Newton s law of collng
More informationBoundary Layer to a System of Viscous Hyperbolic Conservation Laws
Acta Mathematcae Applcatae Snca, Englsh Seres Vol. 24, No. 3 (28) 523 528 DOI: 1.17/s1255-8-861-6 www.applmath.com.cn Acta Mathema ca Applcatae Snca, Englsh Seres The Edtoral Offce of AMAS & Sprnger-Verlag
More informationVector Norms. Chapter 7 Iterative Techniques in Matrix Algebra. Cauchy-Bunyakovsky-Schwarz Inequality for Sums. Distances. Convergence.
Vector Norms Chapter 7 Iteratve Technques n Matrx Algebra Per-Olof Persson persson@berkeley.edu Department of Mathematcs Unversty of Calforna, Berkeley Math 128B Numercal Analyss Defnton A vector norm
More information8/25/17. Data Modeling. Data Modeling. Data Modeling. Patrice Koehl Department of Biological Sciences National University of Singapore
8/5/17 Data Modelng Patrce Koehl Department of Bologcal Scences atonal Unversty of Sngapore http://www.cs.ucdavs.edu/~koehl/teachng/bl59 koehl@cs.ucdavs.edu Data Modelng Ø Data Modelng: least squares Ø
More informationGeometry of Müntz Spaces
WDS'12 Proceedngs of Contrbuted Papers, Part I, 31 35, 212. ISBN 978-8-7378-224-5 MATFYZPRESS Geometry of Müntz Spaces P. Petráček Charles Unversty, Faculty of Mathematcs and Physcs, Prague, Czech Republc.
More informationModelli Clamfim Equazioni differenziali 22 settembre 2016
CLAMFIM Bologna Modell 1 @ Clamfm Equazon dfferenzal 22 settembre 2016 professor Danele Rtell danele.rtell@unbo.t 1/22? Ordnary Dfferental Equatons A dfferental equaton s an equaton that defnes a relatonshp
More informationThe Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
More informationIndeterminate pin-jointed frames (trusses)
Indetermnate pn-jonted frames (trusses) Calculaton of member forces usng force method I. Statcal determnacy. The degree of freedom of any truss can be derved as: w= k d a =, where k s the number of all
More informationBeyond Zudilin s Conjectured q-analog of Schmidt s problem
Beyond Zudln s Conectured q-analog of Schmdt s problem Thotsaporn Ae Thanatpanonda thotsaporn@gmalcom Mathematcs Subect Classfcaton: 11B65 33B99 Abstract Usng the methodology of (rgorous expermental mathematcs
More informationThe Second Anti-Mathima on Game Theory
The Second Ant-Mathma on Game Theory Ath. Kehagas December 1 2006 1 Introducton In ths note we wll examne the noton of game equlbrum for three types of games 1. 2-player 2-acton zero-sum games 2. 2-player
More informationA PROCEDURE FOR SIMULATING THE NONLINEAR CONDUCTION HEAT TRANSFER IN A BODY WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY.
Proceedngs of the th Brazlan Congress of Thermal Scences and Engneerng -- ENCIT 006 Braz. Soc. of Mechancal Scences and Engneerng -- ABCM, Curtba, Brazl,- Dec. 5-8, 006 A PROCEDURE FOR SIMULATING THE NONLINEAR
More informationA New Refinement of Jacobi Method for Solution of Linear System Equations AX=b
Int J Contemp Math Scences, Vol 3, 28, no 17, 819-827 A New Refnement of Jacob Method for Soluton of Lnear System Equatons AX=b F Naem Dafchah Department of Mathematcs, Faculty of Scences Unversty of Gulan,
More informationBallot Paths Avoiding Depth Zero Patterns
Ballot Paths Avodng Depth Zero Patterns Henrch Nederhausen and Shaun Sullvan Florda Atlantc Unversty, Boca Raton, Florda nederha@fauedu, ssull21@fauedu 1 Introducton In a paper by Sapounaks, Tasoulas,
More informationSharp integral inequalities involving high-order partial derivatives. Journal Of Inequalities And Applications, 2008, v. 2008, article no.
Ttle Sharp ntegral nequaltes nvolvng hgh-order partal dervatves Authors Zhao, CJ; Cheung, WS Ctaton Journal Of Inequaltes And Applcatons, 008, v. 008, artcle no. 5747 Issued Date 008 URL http://hdl.handle.net/07/569
More informationGoogle PageRank with Stochastic Matrix
Google PageRank wth Stochastc Matrx Md. Sharq, Puranjt Sanyal, Samk Mtra (M.Sc. Applcatons of Mathematcs) Dscrete Tme Markov Chan Let S be a countable set (usually S s a subset of Z or Z d or R or R d
More informationCOMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
Avalable onlne at http://sck.org J. Math. Comput. Sc. 3 (3), No., 6-3 ISSN: 97-537 COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
More informationON THE BURGERS EQUATION WITH A STOCHASTIC STEPPING STONE NOISY TERM
O THE BURGERS EQUATIO WITH A STOCHASTIC STEPPIG STOE OISY TERM Eaterna T. Kolovsa Comuncacón Técnca o I-2-14/11-7-22 PE/CIMAT On the Burgers Equaton wth a stochastc steppng-stone nosy term Eaterna T. Kolovsa
More informationA Local Variational Problem of Second Order for a Class of Optimal Control Problems with Nonsmooth Objective Function
A Local Varatonal Problem of Second Order for a Class of Optmal Control Problems wth Nonsmooth Objectve Functon Alexander P. Afanasev Insttute for Informaton Transmsson Problems, Russan Academy of Scences,
More informationSL n (F ) Equals its Own Derived Group
Internatonal Journal of Algebra, Vol. 2, 2008, no. 12, 585-594 SL n (F ) Equals ts Own Derved Group Jorge Macel BMCC-The Cty Unversty of New York, CUNY 199 Chambers street, New York, NY 10007, USA macel@cms.nyu.edu
More informationIrregular vibrations in multi-mass discrete-continuous systems torsionally deformed
(2) 4 48 Irregular vbratons n mult-mass dscrete-contnuous systems torsonally deformed Abstract In the paper rregular vbratons of dscrete-contnuous systems consstng of an arbtrary number rgd bodes connected
More informationMemoirs on Dierential Equations and Mathematical Physics Volume 11, 1997, 67{88 Guram Kharatishvili and Tamaz Tadumadze THE PROBLEM OF OPTIMAL CONTROL
Memors on Derental Equatons and Mathematcal Physcs Volume 11, 1997, 67{88 Guram Kharatshvl and Tamaz Tadumadze THE PROBLEM OF OPTIMAL CONTROL FOR NONLINEAR SYSTEMS WITH VARIABLE STRUCTURE, DELAYS AND PIECEWISE
More information4.2 Chemical Driving Force
4.2. CHEMICL DRIVING FORCE 103 4.2 Chemcal Drvng Force second effect of a chemcal concentraton gradent on dffuson s to change the nature of the drvng force. Ths s because dffuson changes the bondng n a
More informationPerron Vectors of an Irreducible Nonnegative Interval Matrix
Perron Vectors of an Irreducble Nonnegatve Interval Matrx Jr Rohn August 4 2005 Abstract As s well known an rreducble nonnegatve matrx possesses a unquely determned Perron vector. As the man result of
More informationMINIMAL MULTI-CONVEX PROJECTIONS ONTO SUBSPACES OF POLYNOMIALS JOANNA MEISSNER 1
MINIMAL MULTI-CONVEX PROJECTIONS ONTO SUBSPACES OF POLYNOMIALS JOANNA MEISSNER IMUJ Preprnt 9/9 Abstract. Let X C N [, ], ), where N 3 and let V be a lnear subspace of Π N, where Π N denotes the space
More informationNUMERICAL EVALUATION OF PERIODIC BOUNDARY CONDITION ON THERMO-MECHANICAL PROBLEM USING HOMOGENIZATION METHOD
THE 9 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS NUMERICAL EVALUATION OF PERIODIC BOUNDAR CONDITION ON THERMO-MECHANICAL PROBLEM USING HOMOGENIZATION METHOD M.R.E. Nasuton *, N. Watanabe, A. Kondo,2
More informationSupplement: Proofs and Technical Details for The Solution Path of the Generalized Lasso
Supplement: Proofs and Techncal Detals for The Soluton Path of the Generalzed Lasso Ryan J. Tbshran Jonathan Taylor In ths document we gve supplementary detals to the paper The Soluton Path of the Generalzed
More informationThe Minimum Universal Cost Flow in an Infeasible Flow Network
Journal of Scences, Islamc Republc of Iran 17(2): 175-180 (2006) Unversty of Tehran, ISSN 1016-1104 http://jscencesutacr The Mnmum Unversal Cost Flow n an Infeasble Flow Network H Saleh Fathabad * M Bagheran
More informationCHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE
CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng
More informationModelli Clamfim Equazione del Calore Lezione ottobre 2014
CLAMFIM Bologna Modell 1 @ Clamfm Equazone del Calore Lezone 17 15 ottobre 2014 professor Danele Rtell danele.rtell@unbo.t 1/24? Convoluton The convoluton of two functons g(t) and f(t) s the functon (g
More informationLecture 3. Ax x i a i. i i
18.409 The Behavor of Algorthms n Practce 2/14/2 Lecturer: Dan Spelman Lecture 3 Scrbe: Arvnd Sankar 1 Largest sngular value In order to bound the condton number, we need an upper bound on the largest
More informationCHEMICAL REACTIONS AND DIFFUSION
CHEMICAL REACTIONS AND DIFFUSION A.K.A. NETWORK THERMODYNAMICS BACKGROUND Classcal thermodynamcs descrbes equlbrum states. Non-equlbrum thermodynamcs descrbes steady states. Network thermodynamcs descrbes
More informationCase Study of Markov Chains Ray-Knight Compactification
Internatonal Journal of Contemporary Mathematcal Scences Vol. 9, 24, no. 6, 753-76 HIKAI Ltd, www.m-har.com http://dx.do.org/.2988/cms.24.46 Case Study of Marov Chans ay-knght Compactfcaton HaXa Du and
More informationMath 217 Fall 2013 Homework 2 Solutions
Math 17 Fall 013 Homework Solutons Due Thursday Sept. 6, 013 5pm Ths homework conssts of 6 problems of 5 ponts each. The total s 30. You need to fully justfy your answer prove that your functon ndeed has
More informationWeak Imposition Of Boundary Conditions For. The Navier-Stokes Equations
Weak Imposton Of Boundary Condtons For The Naver-Stokes Equatons Atfe Çağlar Ph.D. Student Department of Mathematcs Unversty of Pttsburgh Pttsburgh, PA 15260, USA November 23, 2002 Abstract We prove the
More informationSTEINHAUS PROPERTY IN BANACH LATTICES
DEPARTMENT OF MATHEMATICS TECHNICAL REPORT STEINHAUS PROPERTY IN BANACH LATTICES DAMIAN KUBIAK AND DAVID TIDWELL SPRING 2015 No. 2015-1 TENNESSEE TECHNOLOGICAL UNIVERSITY Cookevlle, TN 38505 STEINHAUS
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
More informationFACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP
C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class
More informationResearch Article Relative Smooth Topological Spaces
Advances n Fuzzy Systems Volume 2009, Artcle ID 172917, 5 pages do:10.1155/2009/172917 Research Artcle Relatve Smooth Topologcal Spaces B. Ghazanfar Department of Mathematcs, Faculty of Scence, Lorestan
More informationEEE 241: Linear Systems
EEE : Lnear Systems Summary #: Backpropagaton BACKPROPAGATION The perceptron rule as well as the Wdrow Hoff learnng were desgned to tran sngle layer networks. They suffer from the same dsadvantage: they
More informationVARIATION OF CONSTANT SUM CONSTRAINT FOR INTEGER MODEL WITH NON UNIFORM VARIABLES
VARIATION OF CONSTANT SUM CONSTRAINT FOR INTEGER MODEL WITH NON UNIFORM VARIABLES BÂRZĂ, Slvu Faculty of Mathematcs-Informatcs Spru Haret Unversty barza_slvu@yahoo.com Abstract Ths paper wants to contnue
More informationDimension Reduction and Visualization of the Histogram Data
The 4th Workshop n Symbolc Data Analyss (SDA 214): Tutoral Dmenson Reducton and Vsualzaton of the Hstogram Data Han-Mng Wu ( 吳漢銘 ) Department of Mathematcs Tamkang Unversty Tamsu 25137, Tawan http://www.hmwu.dv.tw
More informationLimit Cycle Generation for Multi-Modal and 2-Dimensional Piecewise Affine Control Systems
Lmt Cycle Generaton for Mult-Modal and 2-Dmensonal Pecewse Affne Control Systems atsuya Ka Department of Appled Electroncs Faculty of Industral Scence and echnology okyo Unversty of Scence 6-3- Njuku Katsushka-ku
More information2E Pattern Recognition Solutions to Introduction to Pattern Recognition, Chapter 2: Bayesian pattern classification
E395 - Pattern Recognton Solutons to Introducton to Pattern Recognton, Chapter : Bayesan pattern classfcaton Preface Ths document s a soluton manual for selected exercses from Introducton to Pattern Recognton
More informationLab session: numerical simulations of sponateous polarization
Lab sesson: numercal smulatons of sponateous polarzaton Emerc Boun & Vncent Calvez CNRS, ENS Lyon, France CIMPA, Hammamet, March 2012 Spontaneous cell polarzaton: the 1D case The Hawkns-Voturez model for
More informationApplication of B-Spline to Numerical Solution of a System of Singularly Perturbed Problems
Mathematca Aeterna, Vol. 1, 011, no. 06, 405 415 Applcaton of B-Splne to Numercal Soluton of a System of Sngularly Perturbed Problems Yogesh Gupta Department of Mathematcs Unted College of Engneerng &
More informationExercise Solutions to Real Analysis
xercse Solutons to Real Analyss Note: References refer to H. L. Royden, Real Analyss xersze 1. Gven any set A any ɛ > 0, there s an open set O such that A O m O m A + ɛ. Soluton 1. If m A =, then there
More informationFixed points of IA-endomorphisms of a free metabelian Lie algebra
Proc. Indan Acad. Sc. (Math. Sc.) Vol. 121, No. 4, November 2011, pp. 405 416. c Indan Academy of Scences Fxed ponts of IA-endomorphsms of a free metabelan Le algebra NAIME EKICI 1 and DEMET PARLAK SÖNMEZ
More informationCOMPLEX NUMBERS AND QUADRATIC EQUATIONS
COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not
More informationIsogeometric Analysis with Geometrically Continuous Functions on Multi-Patch Geometries. Mario Kapl, Florian Buchegger, Michel Bercovier, Bert Jüttler
Isogeometrc Analyss wth Geometrcally Contnuous Functons on Mult-Patch Geometres Maro Kapl Floran Buchegger Mchel Bercover Bert Jüttler G+S Report No 35 August 05 Isogeometrc Analyss wth Geometrcally Contnuous
More informationCIS526: Machine Learning Lecture 3 (Sept 16, 2003) Linear Regression. Preparation help: Xiaoying Huang. x 1 θ 1 output... θ M x M
CIS56: achne Learnng Lecture 3 (Sept 6, 003) Preparaton help: Xaoyng Huang Lnear Regresson Lnear regresson can be represented by a functonal form: f(; θ) = θ 0 0 +θ + + θ = θ = 0 ote: 0 s a dummy attrbute
More informationYong Joon Ryang. 1. Introduction Consider the multicommodity transportation problem with convex quadratic cost function. 1 2 (x x0 ) T Q(x x 0 )
Kangweon-Kyungk Math. Jour. 4 1996), No. 1, pp. 7 16 AN ITERATIVE ROW-ACTION METHOD FOR MULTICOMMODITY TRANSPORTATION PROBLEMS Yong Joon Ryang Abstract. The optmzaton problems wth quadratc constrants often
More information