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

1999
*
Journal of Complexity
*

We are concerned with

doi:10.1006/jcom.1999.0510
fatcat:jtkdqcaqrfeuzpp6ykijzhops4
*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 ...##
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The Problem of Perimeter

2018
*
infocus Magazine
*

Few real world objects have ideal Euclidean

doi:10.22443/rms.inf.1.167
fatcat:zoiscygyvjdlpb7msupv6wpj64
*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 ...##
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On an overdetermined elliptic problem

2011
*
Pacific Journal of Mathematics
*

Hence, in addition to

doi:10.2140/pjm.2011.250.319
fatcat:jom4jvyy4nhknf6nybrwy4s3yy
*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. ...##
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Mathematical problems in transonic flow

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>| . ...

##
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Whitham hierarchy in growth problems

2005
*
Theoretical and mathematical physics
*

This equivalence allows one to distinguish a class of exact solutions to

doi:10.1007/s11232-005-0002-4
fatcat:y7gvwkoobrdb3luz7oejkhne7q
*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. ...##
###
Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem

2013
*
Inverse Problems
*

Some results on

doi:10.1088/0266-5611/29/10/104003
fatcat:gnszfbxkw5dj3geywxyzxlpevi
*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*...##
###
Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci
[article]

2018
*
arXiv
*
pre-print

In this paper we continue our study of bifurcations of solutions of

arXiv:1804.07479v3
fatcat:syeyzld6l5czflcuum5aqhgyce
*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. ...##
###
Vortex design problem

2012
*
Journal of Computational and Applied Mathematics
*

In this investigation we propose a computational approach

doi:10.1016/j.cam.2011.11.002
fatcat:yd4e262wv5afvdc5fn6s2vbb2y
*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 ...##
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Hyperbolic lattice point problems

2011
*
Proceedings of the American Mathematical Society
*

We prove some analogues of planar lattice

doi:10.1090/s0002-9939-2010-10536-1
fatcat:rdxtgprjgbavzmjxasvgqheviq
*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*...##
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The reconstruction problem
[chapter]

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. ...

##
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New multigrid smoothers for the Oseen problem

2010
*
Numerical Linear Algebra with Applications
*

We investigate

doi:10.1002/nla.707
fatcat:vufppcep55gqtgry3xetuhmsb4
*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. ...##
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Hilbert's 16th problem
[article]

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 ...

##
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Scalable computation of thermomechanical turbomachinery problems
[article]

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. ...

##
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TURNING SHAPE DECISION PROBLEMS INTO MEASURES

2004
*
International journal of shape modeling
*

This paper considers

doi:10.1142/s0218654304000614
fatcat:tats7vlypbhblippicfepliwoq
*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 ...##
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A note on some overdetermined elliptic problem
[article]

2010
*
arXiv
*
pre-print

(nonzero) Neumann condtion at

arXiv:1001.1060v1
fatcat:jxb54o4vvfbtvhgz6d4jofzwhu
*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. ...
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