A numerical schema for the transport of nutrients and hormones in plant growth
|
|
- Emil Fields
- 5 years ago
- Views:
Transcription
1 Afr Mat DOI 0007/s z A numerical schema for the transport of nutrients and hormones in plant growth S Bouena A Chiboub J Pousin Received: 24 April 202 / Accepted: 30 November 202 African Mathematical Union and Springer-Verlag Berlin Heidelberg 202 Abstract Classical numerical methods exhibit numerical discrepancies, when we are dealing with the transport equation in domain of heterogeneous sizes In this work, a numerical scheme, based on a domain decomposition strategy is built to avoid numerical discrepancies Let us mention that this work is inspired from the results given in Picq and Pousin (Variational reduction for the transport equation and plants growth, 2007) Keywords Plant growth Branch Two-dimensional domain Finite differences Asymptotic development Variational reduction Algorithms Numerical simulations Introduction The plant growth presents a seasonal periodicity The observation of the growth of the principal branch and formation of new buds show that the distance between them is approximately the same Dynamical models of this phenomena have been studied in [4] wherethe authors had represented a growing plant as a system of intervals, which are called branches The branches appear and grow according to some rules The growing part of the plant contains a narrow exterior part, considered as a surface, where cells proliferate The displacement of this surface corresponds to the plant growth The plant growth is due to the nutrients produced in the roots and transported through the whole of the plant and the appearance of new branches is consequence of the concentration of many hormones The most important of them are cytokinin and Auxin The Auxin is produced in the growing parts of the plant And the S Bouena (B) A Chiboub Sciences Faculty, University Hassan II Casablanca, POB 5366, Maarif, Casablanca, Morocco bouena@yahoofr; sbouena@fsacacma A Chiboub chiboubatika@yahoofr J Pousin Institut Camille Jordan, CNRS UMR 5208, Université Lyon I, Villeurbanne Cedex, France eromepousin@insa-lyonfr 23
2 S Bouena et al Fig A representation of plant as a heterogeneous domain Q Trunk Cytokinin is produced in the roots and in the growing parts of the plant The concentrations of the nutrients and hormones are modelized by convection-diffusion equations In [2] and [3], the one dimensional case without diffusion has been studied, and a numerical scheme has been proposed In this work, the two dimensional case is considered, that is to say, the transport equation is handled in a domain of heterogeneous size Using only a strategy of domain decomposition, the simulation of the nutrients transport by finite differences method exhibits numerical discrepancies in the thin part of the domain representing the branche Thanks to the asymptotic partial decomposition method, we use a perturbation argument in order build an adapted basis to avoid these numerical discrepancies and the transport equation can be reduced to a partial differential equation with less variables in this part of the domain The asymptotic partial decomposition method introduced in [6] isusedinthiswork for the transport equation in an heterogeneous domain with a general right hand A numerical method based on finite elements for such models is given in [5] We show in this work, that the finite differences method works better on the model obtained by the partial asymptotic decomposition strategy than that obtained by the domain decomposition only The article is organized as follows The introduction is ended by recalling the mathematical model In Sect 2, the asymptotic domain decomposition formulation of the problem is analyzed In Sect 3, some numerical results are presented Let us introduce the transport equation in the two-dimensional domain Q which represents a part of the plant (Fig ) Let us give some useful notations Q (0, ) S ( 2 ɛ, 2 + ɛ ) S 2 Q Q 2, where S (0, ) (0,γ)et S 2 ( 2 η, 2 + η) (γ, ) Let f C 0 (Q; R), a(t, x, y) ( a x ) (t, x, y) a y and β a (t, x, y) x (t, x, y) be such that a x C (Q; R + ) and a y C (Q; R + ) a y (t, x, y) In the following, we assume a x and a y to be bounded from below by two positive numbers, and denote by ( / ) the Euclidean inner product Let Q {(t, x, y), (β/n) <0} We look for u H(β, Q) {ρ L 2 (Q), (β/ ρ) L 2 (Q), ρ\ Q L 2 ( Q, (β/n) dσ)} 23
3 A numerical schema for nutrients transport verifying { (β/ u) u t + a x (t, x, y) x u + a y (t, x, y) u y f (t, x, y) in Q () u 0 on Q Theorem The problem () has an unique solution u Proof For a proof the reader is referred to [] 2 The decomposed problem In this section, the problem () is formulated in the context of the partial asymptotic decomposition method First, let us build the vector space of finite dimension well suited for the decomposition of the problem () Here, variable x ( 2 η, 2 + η) is considered as a parameter For a positive regular function ν, we will denote by L 2 (,νdt) the Hilbert s space equipped with the inner product with the weight function ν Lemma Suppose that function a is time independent Let m N and S ɛ ( 2 ɛ, 2 + ɛ) {x} {γ } Define q 0 (t) and q (t) sin( π( t 2 ɛ + )) m Then the functions (q ) 0 m generate M(S ɛ ) a vector space of dimension m + In addition, the functions q : (i) are orthogonal with respect to both the L 2 (S ɛ ), L 2 (S ɛ, a x (, x,γ)) and L 2 (S ɛ, a y (, x,γ))inner product, (ii) verify q k (t)q (t)dt 0 for all 0 k, mwhereq k (t) denotes the derivative function of q k (t) When functions a x, a y are time dependent, we have: Lemma 2 Let m N, suppose a x (,, ) (respectively a y (,, ))tobec and bounded from below by positive constants, then there exists a family of regular functions (w (t)) 0 m for t ( 2 ɛ, 2 + ɛ) which generates M(S ɛ) a vector space of dimension m + In addition, the functions w : (i) are orthogonal with respect to L 2 (S ɛ ), L 2 (S ɛ, a x (, x,γ)dt) and L 2 (S ɛ, a y (, x,γ)dt) inner products, (ii) verify (w k (t))w (t)dt 0 for all 0 k, m, where w k (t) denotes the derivative function of w k (t) Proof Let ( q ) 0 m be the normalized basis derived from the L 2 orthogonal basis (q ) 0 m Since the function a x (,, ) (respectively a y (,, )) is bounded from below by a positive constant, the bilinear form ( q (t))( q k (t))a x (t, x, y) dt (respectively ( q (t))( q k (t))a y (t, x, y)dt)definesanl 2 -inner product Let us denote by M its matrix with respect to the basis ( q ) 0 m The matrix M is symmetric positive 23
4 S Bouena et al defined, thus the spectral theorem claims that there exists an orthogonal basis with respect to L 2 -inner product denoted by (w ) 0 m which diagonalizes the matrix M The expression of the bilinear form in the basis (w ) 0 m is diagonal, thus the vectors (w ) 0 m are orthogonal to each others with respect to the bilinear form Since the (w ) functions are linear combinations of ( q ) functions, the second items of Lemma 2 still holds true for the ) functions Lemma 2 is proved (w We consider, in the following, the general case where the speed depends on time The time-independent case is treated ( by simply replacing the vectors w k by q k a Define the 2-D velocity a x ) (t, x, y) a y and the incoming part of the boundary S (t, x, y) 2 {(x, y), (a/n) <0}, where n is the outward normal to S 2 Definition The decomposed problem associated to the problem () consists in finding (u, u 2 ) such that: (u, u 2 ) H(β, Q ) M(S ɛ ) 0 H (( 2 η, 2 + η) (γ, )), u 2 m w k (t)u 2k (x, y) and (β/ u ) f in Q, k0 u 0 on Q, (β/ u 2 ) f in Q 2, b (u u 2,w k ) 2 (u (t, x,γ) ɛ u 2 (t, x,γ))q k (t)a y (t, x,γ)dt 0, x ( 2 η, 2 + η), q k M(S ɛ ) (2) The last equation of this system is the boundary condition on the incoming part of the domain S 2 in L 2 ( 2 η, 2 + η) Let u be the solution of the problem () andu its restriction on Q We introduce u 2 m k0 w k (t)u 2k (x) Then u 2 is the solution restricted to Q 2 and u 2 M(S ɛ ) 0 H (( 2 η, 2 + η) (γ, )) The decomposed problem is obtained by multiplying () by w k (t) M(S ɛ ) and by integrating with respect to t Observe that u 2 t w k(t) dt 0 w k M(S ɛ ); and obviously the following condition: b(u u 2,w k ) (u (t, x,γ) u 2 (t, x, γ ))w k (t)a y (t, x,γ)dt 0 yields the boundary condition 23
5 A numerical schema for nutrients transport Theorem 2 The decomposed problem given in definition is equivalent to the following reduced problem which consists in finding (u, u 2 ) such that: (u, u 2 ) H(β, Q ) M(S ɛ ) 0 H (( 2 η, 2 + η) (γ, )) and (β/ u ) f in Q, u 0 Q, A x,k u 2 x + Ay,k u 2 y f (t, x, y)w k (t)dt aein S 2, (3) u (t, x,γ)w k (t)a y (t, x,γ)dt u 2k (x,γ) w k(t)w k (t)a y (t, x,γ)dt ae in ( 2 η, 2 + η) and w k M(S ɛ ) where A x,k Proof We have a x (t, x, y)w k (t)w k (t)dt and A y,k u 2 m w k (t)u 2k (x, y) k0 Then almost everywhere in S 2 we have: u 2k x a x (t, x, y)w k (t)w k (t) dt + u 2k y f (t, x, y)w k (t) dt w k M(S ɛ ) a y (t, x, y)w k (t)w k (t)dt a y (t, x, y)w k (t)w k (t) dt Corollary The zero order approximation corresponding to the decomposed problem (3) is given by (β/ u ) 2 f in Q, u 0 Q, a x ( 2, x, y) u 20 x + a y ( 2, x, y) u 20 y f ( 2, x, y) ae in ( 2 η, 2 (4) + η) (γ, ), u ( 2, x,γ) u 20(x,γ) x ( 2 η, 2 + η) 2 Algorithmes for numerical simulation Now, let us come to the numerical approximation of (u, u 2 ), the solution to the decomposed problem (3) To supplement the results outlined in [3], we will give in that follows the construction of the algorithm requiered to the numerical simulations An upwind finite differences method is used for u in Q and for u 2 in Q 2 Let M, N be two fixed positive integers and 0 <ɛ< 2, 0 <η< 2, 0 <γ < be three real numbers 23
6 S Bouena et al 2 The numerical approximation on Q The time and space steps are defined by t N, x M and y γ M Introduce the family of points (t n, x i, y ) such that t n n t, x i i x andy y For all n 0,,,N; i 0,,,M and 0,,,M we pose ui, n u(t n, x i, y ), a x,n a x (t n, x i, y ) and a y,n a y (t n, x i, y ) then we obtain the following schema: u n+ u n t which can be writen as + a x,n+ ui, n+ ui, n+ x A n+ 0 0 B2 n+ A n+ 2 where Θ n+ 0 B 3 n+ A n BM n+ F n+ U n F n+ 2 U n 2 F n+, U n F n+ M + a y,n+ ui, n+ ui, n+ f (t n+, x i, y ) y Θ n+ U n+ U n + F n+ (5) U n M An+ M and F n+ In the other hand the matrices Al n+ ) M for 2 h M are such that (b h,n+ i + o l,n+ x t ax,n+ il + y t a y,n+ il i, i x t ax,n+ il i, 0 elsewhere, U n tf n+ tf n+ 2 tf n+ M U n U2 n U n M, (o l,n+ i ) M for l M and B n+ and b h,n+ i { t y a y,n+ ih h i, 0, elsewhere 22 The numerical approximation on Q 2 For Q 2 the time step is given by ɛ t 2ɛ N and the space steps are defined by ηx 2η M and γ y γ M The nodes are given by x i 2 η + i ηxandy γ + γ y For k 0,,,m and,,m; we put vi, k v k (x i, y ) then, using upwind finite differences method, we can write 23 v k vk i, η x A x,k + vk vk A y,k Fi, k y, (6)
7 A numerical schema for nutrients transport where and A x,k A y,k F k a x (t, x i, y )w k (t)w k (t)dt, a y (t, x i, y )w k (t)w k (t)dt f (t, x i, y )w k (t)dt The integrals A x,k i, A y,k i and Fi, k are computed by using a Gauss quadrature formula Then the schema (6) can be written as B k V k F k such that the matrix B k is defined: C k 0 0 D k 2 Ck 2 B k 0 D3 k Ck D k M Ck M where Cl k (crs l,k ) r,s M and Dl k (drs l,k ) r,s M for l M In the other hand η xa x,k crs l,k r,l + ya y,k r,l rs, { A x,k r,l y rs, and d l,k rs η xa y,k r,l rs, 0 elsewhere 0 elsewhere We denote v k,l V k D v k 2,l Vl k V k k V 0 k + ηx γ yf k, 2 η x γ yf k 2, for l,,m; V k, F k, v k M,l VM k η x γ yfm, k η x γ yf k,l F k η x γ yf k 2,l Fl k F k 2 and F k for l 2,,M η x γ yfm,l k FM k (7) 23
8 S Bouena et al We denote by (U n, V n ) a numerical approximation of the solution (u, u 2 ) of the decomposed problem (3) on whole domain Q at time t n Then the component Ui n of the matrix U n represents an approximation of u (t n, x i, y ) at every node (t n, x i, y ) of the domain Q and the component m k0 w k (t n )vi, k of the matrix V n represents an approximation of u 2 (t n, x i, y ) at every node (t n, x i, y ) of the domain Q 2 The following proposition allows to summarize the previous algorithm which calculates an approximation of the solution (u, u 2 ) of problem (3) on the whole domain Q Proposition The following algorithm gives an approximation of the solution (u, u 2 ) of the decomposed problem (3): in Q U 0 0, t N, x M and y γ M, t n n t for n 0,, 2,, N, x i i x for i 0,, 2,, M, y y for 0,, 2,, M, n+ U n+ U n + F n+ for0 n N, on the inter f ace VM k (x i,γ) u (t,x i,γ )w k (t)a y (t,x i,γ )dt w k(t)w k (t)a y (t,x i,γ )dt for i Mand0 k m, in Q 2 ɛ t 2ɛ N, ηx 2η M and γ y γ M, t n 2 ɛ + n ɛt for n 0,, N, x i 2 η + i ηx for i 0,, M, y γ + γ y for 0,, M, B k V k F k for 0 k m, V m w k (t n )V k for n 0,,, N k0 Conclusion As we can observe in the following simulations considered at final instant, the classical finite differences method doesn t work for thin domains representing branches Besides, the variational reduction method gives a good description of the evolution of concentrations 3 Numerical simulations In what follows, numerical experiments are presented taking a x (t, x, y) a y (t, x, y) 50x + y + exp(0t), f (t, x, y) (2y + x exp(05t))t, γ 2, N 20 and M 200 Only w 0 and w are considered in M(S ɛ ) They are calculated thanks to Gramm-Schmidt process We present in the Fig 2 numerical simulations of the zero order model, the transport equation by finite differences method in the whole domain Q and the model obtained by the variational reduction strategy Different values of the parameters η and ɛ are tested It can be checked that the finite differences method does not give accurate results for too small values of ɛ and η compared to the zero order model which is the reference one In the other hand the proposed variatinal reduction method gives a similar simulation to the zero order model and is not sensitive to the values of these parameters 23
9 A numerical schema for nutrients transport Fig 2 The numerical representation Acknowledgments This work has been supported with a grant PHC Volubilis from the French foreign office and the marocain ministry of education and research MA//246 References Besson, O, Pousin, J: Solution for linear conservation laws with velocity in L Arch Ration Mech Anal 86(), (2007) 2 Bouena, S, Chiboub, A, Pousin, J: Transport equation reduction for a mathematical model in plants growth JMMNP 6(02), (2008) 23
10 S Bouena et al 3 Bouena, S, Chiboub, A, Pousin, J: Variational reduction for the transport equation in a multiple branching plants growth model, Numéro spécial Congrès International JANO 9 JMMNP 5(07), 5 (2007) 4 Bossov, NS, Volpert, V: Dynamic models of plant growth: mathematics and mathematical modeling I Publibook, Paris (2007) 5 Fontevieille, F: Décomposition asymptôtique et éléments finis, thèse de doctorat, université Claude Bernard-Lyon I (2004) 6 Panasenko, GP: Multi-scale Modelling for structures and composites Springer, Berlin (2005) 7 Picq, M, Pousin, J: Variational reduction for the transport equation and plants growth In: Proccedings of the Conference Modelling of the Heterogeneous Materials with Applications in Constructions and Biological Engineering, Czech Technical University, Prague (2007) 23
Data fitting by vector (V,f)-reproducing kernels
Data fitting by vector (V,f-reproducing kernels M-N. Benbourhim to appear in ESAIM.Proc 2007 Abstract In this paper we propose a constructive method to build vector reproducing kernels. We define the notion
More informationA trigonometric orthogonality with respect to a nonnegative Borel measure
Filomat 6:4 01), 689 696 DOI 10.98/FIL104689M Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat A trigonometric orthogonality with
More informationEquilibrium analysis for a mass-conserving model in presence of cavitation
Equilibrium analysis for a mass-conserving model in presence of cavitation Ionel S. Ciuperca CNRS-UMR 58 Université Lyon, MAPLY,, 696 Villeurbanne Cedex, France. e-mail: ciuperca@maply.univ-lyon.fr Mohammed
More informationMoments of the Rudin-Shapiro Polynomials
The Journal of Fourier Analysis and Applications Moments of the Rudin-Shapiro Polynomials Christophe Doche, and Laurent Habsieger Communicated by Hans G. Feichtinger ABSTRACT. We develop a new approach
More informationMonomial transformations of the projective space
Monomial transformations of the projective space Olivier Debarre and Bodo Lass Abstract We prove that, over any field, the dimension of the indeterminacy locus of a rational map f : P n P n defined by
More informationEXISTENCE AND REGULARITY OF SOLUTIONS FOR STOKES SYSTEMS WITH NON-SMOOTH BOUNDARY DATA IN A POLYHEDRON
Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 147, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE AND REGULARITY OF SOLUTIONS FOR
More informationREMARKS ON THE VANISHING OBSTACLE LIMIT FOR A 3D VISCOUS INCOMPRESSIBLE FLUID
REMARKS ON THE VANISHING OBSTACLE LIMIT FOR A 3D VISCOUS INCOMPRESSIBLE FLUID DRAGOŞ IFTIMIE AND JAMES P. KELLIHER Abstract. In [Math. Ann. 336 (2006), 449-489] the authors consider the two dimensional
More informationBanach Journal of Mathematical Analysis ISSN: (electronic)
Banach J. Math. Anal. 6 (2012), no. 1, 139 146 Banach Journal of Mathematical Analysis ISSN: 1735-8787 (electronic) www.emis.de/journals/bjma/ AN EXTENSION OF KY FAN S DOMINANCE THEOREM RAHIM ALIZADEH
More informationSTOKES PROBLEM WITH SEVERAL TYPES OF BOUNDARY CONDITIONS IN AN EXTERIOR DOMAIN
Electronic Journal of Differential Equations, Vol. 2013 2013, No. 196, pp. 1 28. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu STOKES PROBLEM
More informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
More informationCharacterization of half-radial matrices
Characterization of half-radial matrices Iveta Hnětynková, Petr Tichý Faculty of Mathematics and Physics, Charles University, Sokolovská 83, Prague 8, Czech Republic Abstract Numerical radius r(a) is the
More informationParameter Dependent Quasi-Linear Parabolic Equations
CADERNOS DE MATEMÁTICA 4, 39 33 October (23) ARTIGO NÚMERO SMA#79 Parameter Dependent Quasi-Linear Parabolic Equations Cláudia Buttarello Gentile Departamento de Matemática, Universidade Federal de São
More informationMoore-Penrose Inverse of Product Operators in Hilbert C -Modules
Filomat 30:13 (2016), 3397 3402 DOI 10.2298/FIL1613397M Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Moore-Penrose Inverse of
More informationSpectrum and Exact Controllability of a Hybrid System of Elasticity.
Spectrum and Exact Controllability of a Hybrid System of Elasticity. D. Mercier, January 16, 28 Abstract We consider the exact controllability of a hybrid system consisting of an elastic beam, clamped
More informationSome recent results on controllability of coupled parabolic systems: Towards a Kalman condition
Some recent results on controllability of coupled parabolic systems: Towards a Kalman condition F. Ammar Khodja Clermont-Ferrand, June 2011 GOAL: 1 Show the important differences between scalar and non
More informationSOLVABILITY RELATIONS FOR SOME NON FREDHOLM OPERATORS
SOLVABILITY RELATIONS FOR SOME NON FREDHOLM OPERATORS Vitali Vougalter 1, Vitaly Volpert 2 1 Department of Mathematics and Applied Mathematics, University of Cape Town Private Bag, Rondebosch 7701, South
More informationAn LMI description for the cone of Lorentz-positive maps II
An LMI description for the cone of Lorentz-positive maps II Roland Hildebrand October 6, 2008 Abstract Let L n be the n-dimensional second order cone. A linear map from R m to R n is called positive if
More informationHarmonic Polynomials and Dirichlet-Type Problems. 1. Derivatives of x 2 n
Harmonic Polynomials and Dirichlet-Type Problems Sheldon Axler and Wade Ramey 30 May 1995 Abstract. We take a new approach to harmonic polynomials via differentiation. Surprisingly powerful results about
More informationHigh-order ADI schemes for convection-diffusion equations with mixed derivative terms
High-order ADI schemes for convection-diffusion equations with mixed derivative terms B. Düring, M. Fournié and A. Rigal Abstract We consider new high-order Alternating Direction Implicit ADI) schemes
More informationHOMEOMORPHISMS OF BOUNDED VARIATION
HOMEOMORPHISMS OF BOUNDED VARIATION STANISLAV HENCL, PEKKA KOSKELA AND JANI ONNINEN Abstract. We show that the inverse of a planar homeomorphism of bounded variation is also of bounded variation. In higher
More informationOn solving linear systems arising from Shishkin mesh discretizations
On solving linear systems arising from Shishkin mesh discretizations Petr Tichý Faculty of Mathematics and Physics, Charles University joint work with Carlos Echeverría, Jörg Liesen, and Daniel Szyld October
More informationPolarization constant K(n, X) = 1 for entire functions of exponential type
Int. J. Nonlinear Anal. Appl. 6 (2015) No. 2, 35-45 ISSN: 2008-6822 (electronic) http://dx.doi.org/10.22075/ijnaa.2015.252 Polarization constant K(n, X) = 1 for entire functions of exponential type A.
More informationSYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS. Massimo Grosi Filomena Pacella S. L. Yadava. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 21, 2003, 211 226 SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS Massimo Grosi Filomena Pacella S.
More informationGenerating Orthogonal Polynomials and their Derivatives using Vertex Matching-Partitions of Graphs
Generating Orthogonal Polynomials and their Derivatives using Vertex Matching-Partitions of Graphs John P. McSorley, Philip Feinsilver Department of Mathematics Southern Illinois University Carbondale,
More informationStability of an abstract wave equation with delay and a Kelvin Voigt damping
Stability of an abstract wave equation with delay and a Kelvin Voigt damping University of Monastir/UPSAY/LMV-UVSQ Joint work with Serge Nicaise and Cristina Pignotti Outline 1 Problem The idea Stability
More informationInternal Stabilizability of Some Diffusive Models
Journal of Mathematical Analysis and Applications 265, 91 12 (22) doi:1.16/jmaa.21.7694, available online at http://www.idealibrary.com on Internal Stabilizability of Some Diffusive Models Bedr Eddine
More informationSHARP BOUNDARY TRACE INEQUALITIES. 1. Introduction
SHARP BOUNDARY TRACE INEQUALITIES GILES AUCHMUTY Abstract. This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region R N. The inequalities bound (semi-)norms
More informationTitle: Localized self-adjointness of Schrödinger-type operators on Riemannian manifolds. Proposed running head: Schrödinger-type operators on
Title: Localized self-adjointness of Schrödinger-type operators on Riemannian manifolds. Proposed running head: Schrödinger-type operators on manifolds. Author: Ognjen Milatovic Department Address: Department
More informationL p MAXIMAL REGULARITY FOR SECOND ORDER CAUCHY PROBLEMS IS INDEPENDENT OF p
L p MAXIMAL REGULARITY FOR SECOND ORDER CAUCHY PROBLEMS IS INDEPENDENT OF p RALPH CHILL AND SACHI SRIVASTAVA ABSTRACT. If the second order problem ü + B u + Au = f has L p maximal regularity for some p
More informationA Proof of the Focusing Property of Linear Logic
A Proof of the Focusing Property of Linear Logic Olivier LAURENT Laboratoire de l Informatique du Parallélisme (UMR 5668) Université de Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1 46, allée
More informationConnections between connected topological spaces on the set of positive integers
Cent. Eur. J. Math. 11(5) 2013 876-881 DOI: 10.2478/s11533-013-0210-3 Central European Journal of Mathematics Connections between connected topological spaces on the set of positive integers Research Article
More informationHydrodynamic Limits for the Boltzmann Equation
Hydrodynamic Limits for the Boltzmann Equation François Golse Université Paris 7 & Laboratoire J.-L. Lions golse@math.jussieu.fr Academia Sinica, Taipei, December 2004 LECTURE 2 FORMAL INCOMPRESSIBLE HYDRODYNAMIC
More informationLecture I: Asymptotics for large GUE random matrices
Lecture I: Asymptotics for large GUE random matrices Steen Thorbjørnsen, University of Aarhus andom Matrices Definition. Let (Ω, F, P) be a probability space and let n be a positive integer. Then a random
More informationON SOME ELLIPTIC PROBLEMS IN UNBOUNDED DOMAINS
Chin. Ann. Math.??B(?), 200?, 1 20 DOI: 10.1007/s11401-007-0001-x ON SOME ELLIPTIC PROBLEMS IN UNBOUNDED DOMAINS Michel CHIPOT Abstract We present a method allowing to obtain existence of a solution for
More informationON THE DECIMAL EXPANSION OF ALGEBRAIC NUMBERS
Fizikos ir matematikos fakulteto Seminaro darbai, Šiaulių universitetas, 8, 2005, 5 13 ON THE DECIMAL EXPANSION OF ALGEBRAIC NUMBERS Boris ADAMCZEWSKI 1, Yann BUGEAUD 2 1 CNRS, Institut Camille Jordan,
More informationNotes for Elliptic operators
Notes for 18.117 Elliptic operators 1 Differential operators on R n Let U be an open subset of R n and let D k be the differential operator, 1 1 x k. For every multi-index, α = α 1,...,α n, we define A
More informationLinear Algebra II. 7 Inner product spaces. Notes 7 16th December Inner products and orthonormal bases
MTH6140 Linear Algebra II Notes 7 16th December 2010 7 Inner product spaces Ordinary Euclidean space is a 3-dimensional vector space over R, but it is more than that: the extra geometric structure (lengths,
More informationLeft invertible semigroups on Hilbert spaces.
Left invertible semigroups on Hilbert spaces. Hans Zwart Department of Applied Mathematics, Faculty of Electrical Engineering, Mathematics and Computer Science, University of Twente, P.O. Box 217, 75 AE
More informationExistence and uniqueness of solutions for a diffusion model of host parasite dynamics
J. Math. Anal. Appl. 279 (23) 463 474 www.elsevier.com/locate/jmaa Existence and uniqueness of solutions for a diffusion model of host parasite dynamics Michel Langlais a and Fabio Augusto Milner b,,1
More informationDYNAMIC BIFURCATION THEORY OF RAYLEIGH-BÉNARD CONVECTION WITH INFINITE PRANDTL NUMBER
DYNAMIC BIFURCATION THEORY OF RAYLEIGH-BÉNARD CONVECTION WITH INFINITE PRANDTL NUMBER JUNGHO PARK Abstract. We study in this paper the bifurcation and stability of the solutions of the Rayleigh-Bénard
More informationDerivation and Analysis of Piecewise Constant Conservative Approximation for Anisotropic Diffusion Problems
Derivation Analysis of Piecewise Constant Conservative Approximation for Anisotropic Diffusion Problems A. Agouzal, Naïma Debit o cite this version: A. Agouzal, Naïma Debit. Derivation Analysis of Piecewise
More informationBLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED
BLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED TAOUFIK HMIDI AND SAHBI KERAANI Abstract. In this note we prove a refined version of compactness lemma adapted to the blowup analysis
More informationResearch Article The Zeros of Orthogonal Polynomials for Jacobi-Exponential Weights
bstract and pplied nalysis Volume 2012, rticle ID 386359, 17 pages doi:10.1155/2012/386359 Research rticle The Zeros of Orthogonal Polynomials for Jacobi-Exponential Weights Rong Liu and Ying Guang Shi
More informationMixed exterior Laplace s problem
Mixed exterior Laplace s problem Chérif Amrouche, Florian Bonzom Laboratoire de mathématiques appliquées, CNRS UMR 5142, Université de Pau et des Pays de l Adour, IPRA, Avenue de l Université, 64000 Pau
More informationON EQUIVALENCE OF ANALYTIC FUNCTIONS TO RATIONAL REGULAR FUNCTIONS
J. Austral. Math. Soc. (Series A) 43 (1987), 279-286 ON EQUIVALENCE OF ANALYTIC FUNCTIONS TO RATIONAL REGULAR FUNCTIONS WOJC3ECH KUCHARZ (Received 15 April 1986) Communicated by J. H. Rubinstein Abstract
More informationPeter J. Dukes. 22 August, 2012
22 August, 22 Graph decomposition Let G and H be graphs on m n vertices. A decompostion of G into copies of H is a collection {H i } of subgraphs of G such that each H i = H, and every edge of G belongs
More informationJournal of Computational and Applied Mathematics. Multigrid method for solving convection-diffusion problems with dominant convection
Journal of Computational and Applied Mathematics 226 (2009) 77 83 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam
More informationTHE CHARACTERISTIC FUNCTION OF A COMPLEX SYMMETRIC CONTRACTION
THE CHARACTERISTIC FUNCTION OF A COMPLEX SYMMETRIC CONTRACTION NICOLAS CHEVROT, EMMANUEL FRICAIN, AND DAN TIMOTIN Abstract. It is shown that a contraction on a Hilbert space is complex symmetric if and
More informationDisconjugate operators and related differential equations
Disconjugate operators and related differential equations Mariella Cecchi, Zuzana Došlá and Mauro Marini Dedicated to J. Vosmanský on occasion of his 65 th birthday Abstract: There are studied asymptotic
More informationOn Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma. Ben Schweizer 1
On Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma Ben Schweizer 1 January 16, 2017 Abstract: We study connections between four different types of results that
More informationMATH 4030 Differential Geometry Lecture Notes Part 4 last revised on December 4, Elementary tensor calculus
MATH 4030 Differential Geometry Lecture Notes Part 4 last revised on December 4, 205 Elementary tensor calculus We will study in this section some basic multilinear algebra and operations on tensors. Let
More informationAbstract. 1. Introduction
Journal of Computational Mathematics Vol.28, No.2, 2010, 273 288. http://www.global-sci.org/jcm doi:10.4208/jcm.2009.10-m2870 UNIFORM SUPERCONVERGENCE OF GALERKIN METHODS FOR SINGULARLY PERTURBED PROBLEMS
More informationarxiv: v2 [math.dg] 12 Mar 2018
On triangle meshes with valence 6 dominant vertices Jean-Marie Morvan ariv:1802.05851v2 [math.dg] 12 Mar 2018 Abstract We study triangulations T defined on a closed disc satisfying the following condition
More informationMean-Field Limits for Large Particle Systems Lecture 2: From Schrödinger to Hartree
for Large Particle Systems Lecture 2: From Schrödinger to Hartree CMLS, École polytechnique & CNRS, Université Paris-Saclay FRUMAM, Marseilles, March 13-15th 2017 A CRASH COURSE ON QUANTUM N-PARTICLE DYNAMICS
More informationSpace-Time Nonconforming Optimized Schwarz Waveform Relaxation for Heterogeneous Problems and General Geometries
Space-Time Nonconforming Optimized Schwarz Waveform Relaxation for Heterogeneous Problems and General Geometries Laurence Halpern, Caroline Japhet, and Jérémie Szeftel 3 LAGA, Université Paris XIII, Villetaneuse
More informationENERGY DECAY ESTIMATES FOR LIENARD S EQUATION WITH QUADRATIC VISCOUS FEEDBACK
Electronic Journal of Differential Equations, Vol. 00(00, No. 70, pp. 1 1. ISSN: 107-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp ENERGY DECAY ESTIMATES
More informationA method for construction of Lie group invariants
arxiv:1206.4395v1 [math.rt] 20 Jun 2012 A method for construction of Lie group invariants Yu. Palii Laboratory of Information Technologies, Joint Institute for Nuclear Research, Dubna, Russia and Institute
More informationA MIN-MAX THEOREM AND ITS APPLICATIONS TO NONCONSERVATIVE SYSTEMS
IJMMS 3:7, PII. S6738 http://ijmms.hindawi.com Hindawi Publishing Corp. A MIN-MAX THEOREM AND ITS APPLICATIONS TO NONCONSERVATIVE SYSTEMS LI WEIGUO and LI HONGJIE Received August and in revised form 4
More informationNOTE ON FREE CONJUGACY PINCHED ONE-RELATOR GROUPS
NOTE ON FREE CONJUGACY PINCHED ONE-RELATOR GROUPS ABDEREZAK OULD HOUCINE Abstract. (a) Let L be a group having a presentation L = x 1,, x n, y 1,, y m u = v, where u F 1 = x 1,, x n, u 1, v F 2 = y 1,,
More informationSEMILINEAR ELLIPTIC EQUATIONS WITH DEPENDENCE ON THE GRADIENT
Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 139, pp. 1 9. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SEMILINEAR ELLIPTIC
More informationOn the number of cycles in a graph with restricted cycle lengths
On the number of cycles in a graph with restricted cycle lengths Dániel Gerbner, Balázs Keszegh, Cory Palmer, Balázs Patkós arxiv:1610.03476v1 [math.co] 11 Oct 2016 October 12, 2016 Abstract Let L be a
More informationFractal Conservation Laws: Global Smooth Solutions and Vanishing Regularization
Progress in Nonlinear Differential Equations and Their Applications, Vol. 63, 217 224 c 2005 Birkhäuser Verlag Basel/Switzerland Fractal Conservation Laws: Global Smooth Solutions and Vanishing Regularization
More informationTHE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS
THE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS OGNJEN MILATOVIC Abstract. We consider H V = M +V, where (M, g) is a Riemannian manifold (not necessarily
More informationON PATTERNS OCCURRING IN BINARY ALGEBRAIC NUMBERS
ON PATTERNS OCCURRING IN BINARY ALGEBRAIC NUMBERS B. ADAMCZEWSKI AND N. RAMPERSAD Abstract. We prove that every algebraic number contains infinitely many occurrences of 7/3-powers in its binary expansion.
More informationAn angle metric through the notion of Grassmann representative
Electronic Journal of Linear Algebra Volume 18 Volume 18 (009 Article 10 009 An angle metric through the notion of Grassmann representative Grigoris I. Kalogeropoulos gkaloger@math.uoa.gr Athanasios D.
More information1 Last time: least-squares problems
MATH Linear algebra (Fall 07) Lecture Last time: least-squares problems Definition. If A is an m n matrix and b R m, then a least-squares solution to the linear system Ax = b is a vector x R n such that
More informationNew duality operator for complex circulant matrices and a conjecture of Ryser
New duality operator for complex circulant matrices and a conjecture of Ryser Luis H. Gallardo Mathematics University of Brest Brest, France Luis.Gallardo@univ-brest.fr Submitted: May 6, 2015; Accepted:
More informationMatrix Transformations and Statistical Convergence II
Advances in Dynamical Systems and Applications ISSN 0973-532, Volume 6, Number, pp. 7 89 20 http://campus.mst.edu/adsa Matrix Transformations and Statistical Convergence II Bruno de Malafosse LMAH Université
More informationON THE EIGENVALUE OF INFINITE MATRICES WITH NONNEGATIVE OFF-DIAGONAL ELEMENTS
ON THE EIGENVALUE OF INFINITE MATRICES WITH NONNEGATIVE OFF-DIAGONAL ELEMENTS N. APREUTESEI AND V. VOLPERT The paper is devoted to infinite-dimensional difference operators. Some spectral properties of
More informationCharacterizations of some function spaces by the discrete Radon transform on Z n.
Characterizations of some function spaces by the discrete Radon transform on Z n. A. Abouelaz and T. Kawazoe Abstract Let Z n be the lattice in R n and G the set of all discrete hyperplanes in Z n. Similarly
More informationRemarks on Bronštein s root theorem
Remarks on Bronštein s root theorem Guy Métivier January 23, 2017 1 Introduction In [Br1], M.D.Bronštein proved that the roots of hyperbolic polynomials (1.1) p(t, τ) = τ m + m p k (t)τ m k. which depend
More informationComparison of the Rate of Convergence of Various Iterative Methods for the Class of Weak Contractions in Banach Spaces
Thai Journal of Mathematics Volume 11 (2013) Number 11 : 217 226 http://thaijmathincmuacth ISSN 1686-0209 Comparison of the Rate of Convergence of Various Iterative Methods for the Class of Weak Contractions
More informationSufficient conditions for functions to form Riesz bases in L 2 and applications to nonlinear boundary-value problems
Electronic Journal of Differential Equations, Vol. 200(200), No. 74, pp. 0. ISSN: 072-669. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Sufficient conditions
More informationGeneral Mathematics Vol. 16, No. 1 (2008), A. P. Madrid, C. C. Peña
General Mathematics Vol. 16, No. 1 (2008), 41-50 On X - Hadamard and B- derivations 1 A. P. Madrid, C. C. Peña Abstract Let F be an infinite dimensional complex Banach space endowed with a bounded shrinking
More informationarxiv: v1 [math.pr] 22 May 2008
THE LEAST SINGULAR VALUE OF A RANDOM SQUARE MATRIX IS O(n 1/2 ) arxiv:0805.3407v1 [math.pr] 22 May 2008 MARK RUDELSON AND ROMAN VERSHYNIN Abstract. Let A be a matrix whose entries are real i.i.d. centered
More informationALEKSANDROV-TYPE ESTIMATES FOR A PARABOLIC MONGE-AMPÈRE EQUATION
Electronic Journal of Differential Equations, Vol. 2005(2005), No. 11, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) ALEKSANDROV-TYPE
More informationCritical Groups in Saddle Point Theorems without a Finite Dimensional Closed Loop
Math. Nachr. 43 00), 56 64 Critical Groups in Saddle Point Theorems without a Finite Dimensional Closed Loop By Kanishka Perera ) of Florida and Martin Schechter of Irvine Received November 0, 000; accepted
More informationWavelet Frames on the Sphere for Sparse Representations in High Angular Resolution Diusion Imaging. Chen Weiqiang
Wavelet Frames on the Sphere for Sparse Representations in High Angular Resolution Diusion Imaging Chen Weiqiang Overview 1. Introduction to High Angular Resolution Diusion Imaging (HARDI). 2. Wavelets
More informationAtherosclerosis Initiation Modeled as an Inflammatory Process
Math. Model. Nat. Phenom. Vol. 2, No. 2, 2007, pp. 126-141 Atherosclerosis Initiation Modeled as an Inflammatory Process N. El Khatib 1, S. Génieys and V. Volpert Université Lyon 1, Institut Camille Jordan,
More informationIterative Solution of a Matrix Riccati Equation Arising in Stochastic Control
Iterative Solution of a Matrix Riccati Equation Arising in Stochastic Control Chun-Hua Guo Dedicated to Peter Lancaster on the occasion of his 70th birthday We consider iterative methods for finding the
More informationPROPERTIES OF SIMPLE ROOTS
PROPERTIES OF SIMPLE ROOTS ERIC BRODER Let V be a Euclidean space, that is a finite dimensional real linear space with a symmetric positive definite inner product,. Recall that for a root system Δ in V,
More informationTHE LEGENDRE FORMULA IN CLIFFORD ANALYSIS. Guy Laville, Ivan Ramadanoff
Serdica Math. J. 35 (2009), 61 74 THE LEGENDRE FORMULA IN CLIFFORD ANALYSIS Guy Laville, Ivan Ramadanoff Communicated by P. Pflug Abstract. Let R 0,2m+1 be the Clifford algebra of the antieuclidean 2m+1
More informationGlowinski Pironneau method for the 3D ω-ψ equations
280 GUERMOND AND QUARTAPELLE Glowinski Pironneau method for the 3D ω-ψ equations Jean-Luc Guermond and Luigi Quartapelle 1 LIMSI CNRS, Orsay, France, and Dipartimento di Fisica, Politecnico di Milano,
More informationRobustness for a Liouville type theorem in exterior domains
Robustness for a Liouville type theorem in exterior domains Juliette Bouhours 1 arxiv:1207.0329v3 [math.ap] 24 Oct 2014 1 UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris,
More informationSome definitions. Math 1080: Numerical Linear Algebra Chapter 5, Solving Ax = b by Optimization. A-inner product. Important facts
Some definitions Math 1080: Numerical Linear Algebra Chapter 5, Solving Ax = b by Optimization M. M. Sussman sussmanm@math.pitt.edu Office Hours: MW 1:45PM-2:45PM, Thack 622 A matrix A is SPD (Symmetric
More informationSome improvements of Xfem for cracked domains
Some improvements of Xfem for cracked domains E. Chahine 1, P. Laborde 2, J. Pommier 1, Y. Renard 3 and M. Salaün 4 (1) INSA Toulouse, laboratoire MIP, CNRS UMR 5640, Complexe scientifique de Rangueil,
More informationFunctional Analysis HW #5
Functional Analysis HW #5 Sangchul Lee October 29, 2015 Contents 1 Solutions........................................ 1 1 Solutions Exercise 3.4. Show that C([0, 1]) is not a Hilbert space, that is, there
More informationFunctions with orthogonal Hessian
Functions with orthogonal Hessian B. Dacorogna P. Marcellini E. Paolini Abstract A Dirichlet problem for orthogonal Hessians in two dimensions is eplicitly solved, by characterizing all piecewise C 2 functions
More informationNear-Isometry by Relaxation: Supplement
000 00 00 003 004 005 006 007 008 009 00 0 0 03 04 05 06 07 08 09 00 0 0 03 04 05 06 07 08 09 030 03 03 033 034 035 036 037 038 039 040 04 04 043 044 045 046 047 048 049 050 05 05 053 Near-Isometry by
More informationAlmost Invariant Half-Spaces of Operators on Banach Spaces
Integr. equ. oper. theory Online First c 2009 Birkhäuser Verlag Basel/Switzerland DOI 10.1007/s00020-009-1708-8 Integral Equations and Operator Theory Almost Invariant Half-Spaces of Operators on Banach
More informationNew phenomena for the null controllability of parabolic systems: Minim
New phenomena for the null controllability of parabolic systems F.Ammar Khodja, M. González-Burgos & L. de Teresa Aix-Marseille Université, CNRS, Centrale Marseille, l2m, UMR 7373, Marseille, France assia.benabdallah@univ-amu.fr
More informationCANONICAL LOSSLESS STATE-SPACE SYSTEMS: STAIRCASE FORMS AND THE SCHUR ALGORITHM
CANONICAL LOSSLESS STATE-SPACE SYSTEMS: STAIRCASE FORMS AND THE SCHUR ALGORITHM Ralf L.M. Peeters Bernard Hanzon Martine Olivi Dept. Mathematics, Universiteit Maastricht, P.O. Box 616, 6200 MD Maastricht,
More informationMULTIPLICATIVE FIBRE MAPS
MULTIPLICATIVE FIBRE MAPS BY LARRY SMITH 1 Communicated by John Milnor, January 9, 1967 In this note we shall outline a result concerning the cohomology of a multiplicative fibre map. To fix our notation
More informationarxiv: v1 [math.na] 27 Jan 2016
Virtual Element Method for fourth order problems: L 2 estimates Claudia Chinosi a, L. Donatella Marini b arxiv:1601.07484v1 [math.na] 27 Jan 2016 a Dipartimento di Scienze e Innovazione Tecnologica, Università
More informationhere, this space is in fact infinite-dimensional, so t σ ess. Exercise Let T B(H) be a self-adjoint operator on an infinitedimensional
15. Perturbations by compact operators In this chapter, we study the stability (or lack thereof) of various spectral properties under small perturbations. Here s the type of situation we have in mind:
More informationRiemannian geometry of surfaces
Riemannian geometry of surfaces In this note, we will learn how to make sense of the concepts of differential geometry on a surface M, which is not necessarily situated in R 3. This intrinsic approach
More informationLORENZO BRANDOLESE AND JIAO HE
UNIQUENESS THEOREMS FOR THE BOUSSINESQ SYSTEM LORENZO BRANDOLESE AND JIAO HE Abstract. We address the uniqueness problem for mild solutions of the Boussinesq system in R 3. We provide several uniqueness
More informationRegularity for the optimal transportation problem with Euclidean distance squared cost on the embedded sphere
Regularity for the optimal transportation problem with Euclidean distance squared cost on the embedded sphere Jun Kitagawa and Micah Warren January 6, 011 Abstract We give a sufficient condition on initial
More informationPerturbation Theory for Self-Adjoint Operators in Krein spaces
Perturbation Theory for Self-Adjoint Operators in Krein spaces Carsten Trunk Institut für Mathematik, Technische Universität Ilmenau, Postfach 10 05 65, 98684 Ilmenau, Germany E-mail: carsten.trunk@tu-ilmenau.de
More informationNON POSITIVELY CURVED METRIC IN THE SPACE OF POSITIVE DEFINITE INFINITE MATRICES
REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Volumen 48, Número 1, 2007, Páginas 7 15 NON POSITIVELY CURVED METRIC IN THE SPACE OF POSITIVE DEFINITE INFINITE MATRICES ESTEBAN ANDRUCHOW AND ALEJANDRO VARELA
More information