scholarly journals Formal oscillatory distributions

2021 ◽  
pp. 1-19
Author(s):  
Alexander Karabegov
Keyword(s):  
Asymptotic Expansion ◽  
Critical Point ◽  
Phase Function ◽  
Star Product ◽  
Oscillatory Integral ◽  
Poisson Manifold ◽  
Formal Asymptotic Expansion ◽  
Formal Distribution ◽  
Nondegenerate Critical Point

The formal asymptotic expansion of an oscillatory integral whose phase function has one nondegenerate critical point is a formal distribution supported at the critical point which is applied to the amplitude. This formal distribution is called a formal oscillatory integral (FOI). We introduce the notion of a formal oscillatory distribution supported at a point. We prove that a formal distribution is given by some FOI if and only if it is an oscillatory distribution that has a certain nondegeneracy property. We also prove that a star product ⋆ on a Poisson manifold M is natural in the sense of Gutt and Rawnsley if and only if the formal distribution f ⊗ g ↦ ( f ⋆ g ) ( x ) is oscillatory for every x ∈ M.

scholarly journals On the Level Set of a Function with Degenerate Minimum Point

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Yasuhiko Kamiyama
Keyword(s):  
Critical Point ◽  
Level Set ◽  
Smooth Function ◽  
Closed Manifold ◽  
Minimum Point ◽  
Unique Point ◽  
Degenerate Minimum ◽  
Nondegenerate Critical Point ◽  
Smooth Closed Manifold

Forn≥2, letMbe ann-dimensional smooth closed manifold andf:M→Ra smooth function. We setminf(M)=mand assume thatmis attained by unique pointp∈Msuch thatpis a nondegenerate critical point. Then the Morse lemma tells us that ifais slightly bigger thanm,f-1(a)is diffeomorphic toSn-1. In this paper, we relax the condition onpfrom being nondegenerate to being an isolated critical point and obtain the same consequence. Some application to the topology of polygon spaces is also included.

scholarly journals Critical point behaviour of the diffusion length for radiative transfer

1985 ◽  
Vol 27 (2) ◽  
pp. 245-257
Author(s):  
I. F. Grant ◽  
B. H. J. McKellar
Keyword(s):  
Radiative Transfer ◽  
Critical Point ◽  
Infinite Series ◽  
Phase Function ◽  
Transfer Equation ◽  
Continued Fraction ◽  
Diffusion Length ◽  
Radiative Transfer Equation ◽  
Homogenous Medium ◽  
Legendre Expansion

AbstractCritical point behaviour of the diffusion length γ for the solutions of the radiative transfer equation deep in a homogenous medium is studied. The Legendre expansion of the medium's phase function P(cos ψ) is taken to be an infinite series and is characterized by the parameters h0, h1h2,…. A characteristic equation for γ is given in terms of an infinite continued fraction. From this equation it is shown that as any one of the hn, say hp, approaches zero, the others being held constant, γ behaves as , where the critical exponent is found to be vp = ½ for all p = 0, 1, 2,….

scholarly journals A generalisation of the Morse inequalities

2001 ◽  
Vol 70 (3) ◽  
pp. 351-386
Author(s):  
Mohan Bhupal
Keyword(s):  
Critical Point ◽  
Maslov Index ◽  
Closed Manifold ◽  
Zero Section ◽  
Second Variation ◽  
Action Functional ◽  
Jet Bundle ◽  
Morse Inequalities ◽  
The Second Variation ◽  
Nondegenerate Critical Point

AbstractIn this paper we construct a family of variational families for a Legendrian embedding, into the 1-jet bundle of a closed manifold, that can be obtained from the zero section through Legendrian embdeddings, by discretising the action functional. We compute the second variation of a generating funciton obtained as above at a nondegenerate critical point and prove a formula relating the signature of the second variation to the Maslov index as the mesh goes to zero. We use this to prove a generlisation of the Morse inequalities thus refining a theorem of Chekanov.

scholarly journals Single blow-up solutions for a slightly subcritical biharmonic equation

2006 ◽  
Vol 2006 ◽  
pp. 1-20 ◽  
Author(s):  
Khalil El Mehdi
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Critical Exponent ◽  
Critical Point ◽  
Bounded Domain ◽  
Biharmonic Equation ◽  
Blow Up ◽  
Asymptotic Behavior Of Solutions ◽  
Smooth Bounded Domain ◽  
Nondegenerate Critical Point

We consider a biharmonic equation under the Navier boundary condition and with a nearly critical exponent (Pε):∆2u=u9−ε,u>0inΩandu=∆u=0on∂Ω, whereΩis a smooth bounded domain inℝ5,ε>0. We study the asymptotic behavior of solutions of (Pε) which are minimizing for the Sobolev quotient asεgoes to zero. We show that such solutions concentrate around a pointx0∈Ωasε→0, moreoverx0is a critical point of the Robin's function. Conversely, we show that for any nondegenerate critical pointx0of the Robin's function, there exist solutions of (Pε) concentrating aroundx0asε→0.

scholarly journals Asymptotic expansions for distributions of sums of a double sequence of quasilattice random variables

2011 ◽  
Vol 52 ◽  
pp. 353-358
Author(s):  
Algimantas Bikelis ◽  
Juozas Augutis ◽  
Kazimieras Padvelskis
Keyword(s):  
Asymptotic Expansion ◽  
Probability Distribution ◽  
Asymptotic Expansions ◽  
Probability Distributions ◽  
Double Sequence ◽  
Random Variables ◽  
Independent Random Variables ◽  
Infinitely Divisible ◽  
Formal Asymptotic Expansion

We consider the formal asymptotic expansion of probability distribution of the sums of independent random variables. The approximation was made by using infinitely divisible probability distributions.  

scholarly journals ON SINGULAR SOLUTIONS OF THE STATIONARY NAVIER-STOKES SYSTEM IN POWER CUSP DOMAINS

2021 ◽  
Vol 26 (4) ◽  
pp. 651-668
Author(s):  
Konstantinas Pileckas ◽  
Alicija Raciene
Keyword(s):  
Asymptotic Expansion ◽  
Stokes System ◽  
Boundary Value ◽  
Singular Solutions ◽  
Navier Stokes ◽  
Asymptotic Decomposition ◽  
Dirichlet Integral ◽  
Existence Of A Solution ◽  
Source Sink ◽  
Formal Asymptotic Expansion

The boundary value problem for the steady Navier–Stokes system is considered in a 2D bounded domain with the boundary having a power cusp singularity at the point O. The case of a boundary value with a nonzero flow rate is studied. In this case there is a source/sink in O and the solution necessarily has an infinite Dirichlet integral. The formal asymptotic expansion of the solution near the singular point is constructed and the existence of a solution having this asymptotic decomposition is proved.

scholarly journals On deformation of foliations with a center in the projective space

2001 ◽  
Vol 73 (2) ◽  
pp. 191-196 ◽  
Author(s):  
HOSSEIN MOVASATI
Keyword(s):  
Lower Bound ◽  
Projective Space ◽  
Critical Point ◽  
Limit Cycles ◽  
First Integral ◽  
Real Plane ◽  
Real Polynomial ◽  
Fixed Degree ◽  
Polynomial Differential Equations ◽  
Nondegenerate Critical Point

Let <img ALIGN="BOTTOM" src="http:/img/fbpe/aabc/v73n2/m4img1.gif"> be a foliation in the projective space of dimension two with a first integral of the type <img ALIGN="MIDDLE" src="http:/img/fbpe/aabc/v73n2/m4img2.gif">, where F and G are two polynomials on an affine coordinate, <img ALIGN="MIDDLE" src="http:/img/fbpe/aabc/v73n2/m4img3.gif"> = <img ALIGN="MIDDLE" src="http:/img/fbpe/aabc/v73n2/m4img4.gif"> and g.c.d.(p, q) = 1. Let z be a nondegenerate critical point of <img ALIGN="MIDDLE" src="http:/img/fbpe/aabc/v73n2/m4img2.gif">, which is a center singularity of <img ALIGN="BOTTOM" src="http:/img/fbpe/aabc/v73n2/m4img1.gif">, and <img src="http:/img/fbpe/aabc/v73n2/ft.gif" alt="ft.gif (149 bytes)" align="middle"> be a deformation of <img ALIGN="BOTTOM" src="http:/img/fbpe/aabc/v73n2/m4img1.gif"> in the space of foliations of degree deg(<img ALIGN="BOTTOM" src="http:/img/fbpe/aabc/v73n2/m4img1.gif">) such that its unique deformed singularity <img src="http:/img/fbpe/aabc/v73n2/zt.gif" alt="zt.gif (118 bytes)"> near z persists in being a center. We will prove that the foliation <img src="http:/img/fbpe/aabc/v73n2/ft.gif" alt="ft.gif (149 bytes)" align="middle"> has a first integral of the same type of <img ALIGN="BOTTOM" src="http:/img/fbpe/aabc/v73n2/m4img1.gif">. Using the arguments of the proof of this result we will give a lower bound for the maximum number of limit cycles of real polynomial differential equations of a fixed degree in the real plane.

Uniform asymptotic expansion of an integral with a saddle point, a pole and a branch point

1989 ◽  
Vol 426 (1871) ◽  
pp. 273-286 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Saddle Point ◽  
Asymptotic Behaviour ◽  
Critical Point ◽  
Branch Point ◽  
Critical Points ◽  
Uniform Asymptotic Expansion ◽  
Uniform Expansions ◽  
Uniform Expansion ◽  
Special Case

The paper is concerned with an asymptotic behaviour of a contour integral with three critical points: a saddle point, a pole and a branch point. Two leading terms of a uniform asymptotic expansion of the integral have been effectively constructed. The expansion remains valid as its critical points approach one another or coalesce. In the special case that the saddle point is bounded away from one of the remaining critical points, or from both, the expansion reduces to simpler in form quasi-uniform and non-uniform expansions, respectively. With the aid of the non-uniform expansion the integral has been interpreted as describing three different waves, each one associated with a corresponding critical point.

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