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Where Does Smoothness Count the Most for Two-Point Boundary-Value Problems?

Arthur G Werschulz
1999 Journal of Complexity  
We are concerned with the complexity of 2mth order elliptic two-point boundary-value problems Lu = f.  ...  In this paper, we study problems in which f has r derivatives in the L p -sense and for L having the usual divergence form  ...  Some general remarks Recall that the goal of this paper is to nd the complexity of our class of two-point boundary-value problems, as well as optimal algorithms, for the classes of continuous and standard  ... 
doi:10.1006/jcom.1999.0510 fatcat:jtkdqcaqrfeuzpp6ykijzhops4

The Problem of Perimeter

John Russ
2018 infocus Magazine  
Few real world objects have ideal Euclidean boundaries; most reveal more and more irregularities as magnification increases.  ...  When an image of the object is digitised, it becomes necessary to decide how to define the boundary and the pixels that determine it, and how to measure the boundary length.  ...  Measuring the area by counting pixels gives a result for the hole of 94, which is the average of the areas within the two outlines, but the average of the two line lengths, or the length of a line midway  ... 
doi:10.22443/rms.inf.1.167 fatcat:zoiscygyvjdlpb7msupv6wpj64

On an overdetermined elliptic problem

Laurent Hauswirth, Frédéric Hélein, Frank Pacard
2011 Pacific Journal of Mathematics  
Hence, in addition to the regular boundary F(i ‫,)ޒ‬ which is a smooth periodic curve, the image of F has a singular boundary: the set of limits of F(x + i y) as u tends to +∞, for the values of y for  ...  Our results raise the problem of the classification of (unbounded) smooth mdimensional exceptional manifolds.  ... 
doi:10.2140/pjm.2011.250.319 fatcat:jom4jvyy4nhknf6nybrwy4s3yy

Mathematical problems in transonic flow

Cathleen Synge Morawetz
1986 Canadian mathematical bulletin  
The problem is simplified and the important role of modifying the equations with physics terms is examined.  ...  We present an outline of the problem of irrotational compressible flow past an airfoil at speeds that lie somewhere between those of the supersonic flight of the Concorde and the subsonic flight of commercial  ...  Of course adding v smooths out the solution and the solution is found as the fixed point of a mapping using say the Schauder theorem. But there does not appear to be any control on the max | V(j>| .  ... 
doi:10.4153/cmb-1986-023-3 fatcat:7do5omn2ozhzxf2jbhyb3jm6jy

Whitham hierarchy in growth problems

A. V. Zabrodin
2005 Theoretical and mathematical physics  
This equivalence allows one to distinguish a class of exact solutions to the Laplacian growth problem in the multiply-connected case.  ...  We discuss the recently established equivalence between the Laplacian growth in the limit of zero surface tension and the universal Whitham hierarchy known in soliton theory.  ...  Acknowledgments The author is grateful to O.Agam, E.Bettelheim, I.Krichever, A.Marshakov, M.Mineev-Weinstein, R.Theodorescu and P.Wiegmann for collaboration and useful discussions.  ... 
doi:10.1007/s11232-005-0002-4 fatcat:y7gvwkoobrdb3luz7oejkhne7q

Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem

E Lakshtanov, B Vainberg
2013 Inverse Problems  
Some results on the discreteness and localization of the spectrum are obtained for complex valued n(x).  ...  All the results are obtained under the assumption that n(x)-1 does not vanish at the boundary of the obstacle or it vanishes identically, but its normal derivative does not vanish at the boundary.  ...  Grushin [21] , [22] , who calculated the full symbols of different pseudo-differential operators that map boundary values of one problem for an elliptic equation to the boundary values of another problem  ... 
doi:10.1088/0266-5611/29/10/104003 fatcat:gnszfbxkw5dj3geywxyzxlpevi

Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci [article]

Robert I McLachlan, Christian Offen
2018 arXiv   pre-print
In this paper we continue our study of bifurcations of solutions of boundary-value problems for symplectic maps arising as Hamiltonian diffeomorphisms.  ...  The framework is then used to prove the existence of obstructions arising from conformal symplectic symmetries on the bifurcation behaviour of solutions to Hamiltonian boundary value problems.  ...  Acknowledgements We thank Peter Donelan, Bernd Krauskopf, Hinke Osinga and Gemma Mason for useful discussions. This research was supported by the Marsden Fund of the Royal Society Te Apārangi.  ... 
arXiv:1804.07479v3 fatcat:syeyzld6l5czflcuum5aqhgyce

Vortex design problem

Bartosz Protas
2012 Journal of Computational and Applied Mathematics  
In this investigation we propose a computational approach for the solution of optimal control problems for vortex systems with compactly supported vorticity.  ...  Recognizing such Euler flows as free-boundary problems, the proposed approach relies on shape differentiation combined with adjoint analysis to determine cost functional gradients.  ...  Acknowledgments The author wishes to thank Alan Elcrat for interesting discussions, and is also grateful to Professors Alan Elcrat and Ken Miller for providing him with the code for the solution of Euler  ... 
doi:10.1016/j.cam.2011.11.002 fatcat:yd4e262wv5afvdc5fn6s2vbb2y

Hyperbolic lattice point problems

Fernando Chamizo
2011 Proceedings of the American Mathematical Society  
We prove some analogues of planar lattice point problems replacing R 2 by the Poincaré model of the hyperbolic plane and using the orbit of a point under the modular group instead of the lattice generated  ...  Acknowledgments The author is grateful to the mathematics department of Rutgers University for providing hospitality while this work was completed. The author also would like to thank E.  ...  This is in fact a fundamental problem because the classic planar lattice point problem (approximate the number of lattice points in enlarging convex regions [6] ) is reduced to it after dividing the boundary  ... 
doi:10.1090/s0002-9939-2010-10536-1 fatcat:rdxtgprjgbavzmjxasvgqheviq

The reconstruction problem [chapter]

William Lionheart, Nicholas Polydorides, Andrea Borsic
2004 Series in Medical Physics and Biomedical Engineering  
For a given set of test functions v (1.45) is the weak formulation of the boundary value problem for 1.1 with current density specified on the electrodes.  ...  This mixed boundary value problem is well-posed, and the resulting currents are I l = E l σ∂φ/∂n.  ... 
doi:10.1201/9781420034462.pt1 fatcat:zzwaarjx6jez7b7bmv267r654y

New multigrid smoothers for the Oseen problem

Steven Hamilton, Michele Benzi, Eldad Haber
2010 Numerical Linear Algebra with Applications  
We investigate the performance of smoothers based on the Hermitian/skew-Hermitian (HSS) and augmented Lagrangian (AL) splittings applied to MAC discretizations of the Oseen problem.  ...  Local Fourier analysis and numerical experiments on a 2-D lid-driven cavity problem indicate that the proposed smoothers result in hindependent convergence and are fairly robust with respect to the Reynolds  ...  ν; see also [15] where the difficulty of smoothing for the Navier-Stokes equations with low viscosity is pointed out.  ... 
doi:10.1002/nla.707 fatcat:vufppcep55gqtgry3xetuhmsb4

Hilbert's 16th problem [article]

Pablo Pedregal
2021 arXiv   pre-print
The strategy of proof brings variational techniques into the differential-system field by transforming the task of counting limit cycles into counting critical points for a certain smooth, non-negative  ...  We thus solve the second part of Hilbert's 16th problem providing a uniform upper bound for the number of limit cycles which only depends on the degree of the polynomial differential system.  ...  A subset S ⊂ H is said to be a valley for E if it is simply connected, the boundary ∂S does not contain any of the critical points of E, and S is invariant under X(t, ·) for every small positive t, i.e  ... 
arXiv:2103.07193v1 fatcat:k2bm7zkac5eg3fx3t2r5ndp3ga

Scalable computation of thermomechanical turbomachinery problems [article]

Chris N. Richardson, Nathan Sime, Garth N. Wells
2018 arXiv   pre-print
problems and for other physical processes, such as contact, that are important in turbomachinery analysis.  ...  The performance demonstrated in this work opens up the possibility of system-level thermomechanical modelling, and lays the foundation for further research into high-performance formulations for even larger  ...  Acknowledgements We thank Dr Mark Adams of Lawrence Berkeley National Laboratory for his advice on the use of smoothed aggregation AMG.  ... 
arXiv:1804.10060v1 fatcat:2c4sbmv7nne37iyabqunx2u62a

TURNING SHAPE DECISION PROBLEMS INTO MEASURES

RALPH R. MARTIN, PAUL L. ROSIN
2004 International journal of shape modeling  
This paper considers the problem of constructing shape measures; we start by giving a short overview of areas of practical application of such measures.  ...  We give two general principles for constructing measures in this way, and show how they can be applied to construct various shape measures, including ones for convexity, circularity, ellipticity, triangularity  ...  Acknowledgments We would like to thank the members of the Geometric Modelling Society for various helpful discussions, including suggesting some shape properties for measurement, and pointers to references  ... 
doi:10.1142/s0218654304000614 fatcat:tats7vlypbhblippicfepliwoq

A note on some overdetermined elliptic problem [article]

Frédéric Hélein, Frank Pacard
2010 arXiv   pre-print
(nonzero) Neumann condtion at the boundary.  ...  We study the two-dimensional case: we present various examples and give a general construction algorithm of such surface by using complex analysis.  ...  Acknowledgments : We thank Romain Dujardin and Charles Favre for useful discussions. The third author is also partially supported by the ANR-08-BLANC-0335-01 grant.  ... 
arXiv:1001.1060v1 fatcat:jxb54o4vvfbtvhgz6d4jofzwhu
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