Search results for "asymptotic expansion"
showing 10 items of 38 documents
Complex powers of elliptic pseudodifferential operators
1986
The aim of this paper is the construction of complex powers of elliptic pseudodifferential operators and the study of the analytic properties of the corresponding kernels kS (x,y). For x=y, the case of principal interest, the domain of holomorphy and the singularities of kS (x,x) are shown to depend on the asymptotic expansion of the symbol. For classical symbols, kS (x,x) is known to be meromorphic on ℂ with simple poles in a set of equidistant points on the real axis. In the more general cases considered here, the singularities may be distributed over a half plane and kS (x,x) can not always be extended to337-2. An example is given where kS (x,x) has a vertical line as natural boundary.
Zero Viscosity Limit for Analytic Solutions of the Primitive Equations
2016
The aim of this paper is to prove that the solutions of the primitive equations converge, in the zero viscosity limit, to the solutions of the hydrostatic Euler equations. We construct the solution of the primitive equations through a matched asymptotic expansion involving the solution of the hydrostatic Euler equation and boundary layer correctors as the first order term, and an error that we show to be \({O(\sqrt{\nu})}\). The main assumption is spatial analyticity of the initial datum.
Almost Planar Homoclinic Loops in R3
1996
AbstractIn this paper we study homoclinic loops of vector fields in 3-dimensional space when the two principal eigenvalues are real of opposite sign, which we call almost planar. We are interested to have a theory for higher codimension bifurcations. Almost planar homoclinic loop bifurcations generically occur in two versions “non-twisted” and “twisted” loops. We consider high codimension homoclinic loop bifurcations under generic conditions. The generic condition forces the existence of a 2-dimensional topological invariant ring (non necessarily unique), which is a topological cylinder in the “non-twisted” case and a topological Möbius band in the “twisted” case. If the third eigenvalue is…
Asymptotic stability of solutions to Volterra-renewal integral equations with space maps
2012
Abstract In this paper we consider linear Volterra-renewal integral equations (VIEs) whose solutions depend on a space variable, via a map transformation. We investigate the asymptotic properties of the solutions, and study the asymptotic stability of a numerical method based on direct quadrature in time and interpolation in space. We show its properties through test examples.
A Dirichlet problem for the Laplace operator in a domain with a small hole close to the boundary
2016
We study the Dirichlet problem in a domain with a small hole close to the boundary. To do so, for each pair $\boldsymbol\varepsilon = (\varepsilon_1, \varepsilon_2 )$ of positive parameters, we consider a perforated domain $\Omega_{\boldsymbol\varepsilon}$ obtained by making a small hole of size $\varepsilon_1 \varepsilon_2 $ in an open regular subset $\Omega$ of $\mathbb{R}^n$ at distance $\varepsilon_1$ from the boundary $\partial\Omega$. As $\varepsilon_1 \to 0$, the perforation shrinks to a point and, at the same time, approaches the boundary. When $\boldsymbol\varepsilon \to (0,0)$, the size of the hole shrinks at a faster rate than its approach to the boundary. We denote by $u_{\bolds…
Application of the star-product method to the angular momentum quantization
1992
We define a *-product on ℝ3 and solve the polarization equation f*C=0 where C is the Casimir of the coadjoint representation of SO(3). We compute the action of SO(3) on the space of solutions. We then examine the case of non-zero eigenvalues of C, in order to find finite-dimensional representations of SO(3). Finally, we compute \(\sqrt C *\sqrt C \) as an asymptotic series of C. This gives an explanation of the use of the star square root of C in a paper by Bayen et al. instead of its natural square root.
Solutions of nonlinear PDEs in the sense of averages
2012
Abstract We characterize p-harmonic functions including p = 1 and p = ∞ by using mean value properties extending classical results of Privaloff from the linear case p = 2 to all pʼs. We describe a class of random tug-of-war games whose value functions approach p-harmonic functions as the step goes to zero for the full range 1 p ∞ .
On lacunary Toeplitz determinants
2014
By using Riemann--Hilbert problem based techniques, we obtain the asymptotic expansion of lacunary Toeplitz determinants $\det_N\big[ c_{\ell_a-m_b}[f] \big]$ generated by holomorhpic symbols, where $\ell_a=a$ (resp. $m_b=b$) except for a finite subset of indices $a=h_1,\dots, h_n$ (resp. $b=t_1,\dots, t_r$). In addition to the usual Szeg\"{o} asymptotics, our answer involves a determinant of size $n+r$.
Identification of small inhomogeneities: Asymptotic factorization
2007
We consider the boundary value problem of calculating the electrostatic potential for a homogeneous conductor containing finitely many small insulating inclusions. We give a new proof of the asymptotic expansion of the electrostatic potential in terms of the background potential, the location of the inhomogeneities and their geometry, as the size of the inhomogeneities tends to zero. Such asymptotic expansions have already been used to design direct (i.e. noniterative) reconstruction algorithms for the determination of the location of the small inclusions from electrostatic measurements on the boundary, e.g. MUSIC-type methods. Our derivation of the asymptotic formulas is based on integral …
On the time function of the Dulac map for families of meromorphic vector fields
2003
Given an analytic family of vector fields in Bbb R2 having a saddle point, we study the asymptotic development of the time function along the union of the two separatrices. We obtain a result (depending uniformly on the parameters) which we apply to investigate the bifurcation of critical periods of quadratic centres.