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 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.

Scaling Near the Critical Point in Mixture

2005 ◽  
Author(s):  
Eldred H. Chimowitz
Keyword(s):  
Critical Point ◽  
Binary Systems ◽  
Critical Line ◽  
Phase Region ◽  
Unique Point ◽  
Pure Fluid ◽  
High Volatility ◽  
Two Phases ◽  
The One ◽  
Dew Line

The critical point of mixtures requires a more intricate set of conditions to hold than those at a pure-fluid critical point. In contrast to the pure-fluid case, in which the critical point occurs at a unique point, mixtures have additional thermodynamic degrees of freedom. They, therefore, possess a critical line which defines a locus of critical points for the mixture. At each point along this locus, the mixture exhibits a critical point with its own composition, temperature, and pressure. In this chapter we investigate the critical behavior of binary mixtures, since higher-order systems do not bring significant new considerations beyond those found in binaries. We deal first with mixtures at finite compositions along the critical locus, followed by consideration of the technologically important case involving dilute mixtures near the solvent’s critical point. Before taking up this discussion, however, we briefly describe some of the main topographic features of the critical line of systems of significant interest: those for which nonvolatile solutes are dissolved in a solvent near its critical point. The critical line divides the P–T plane into two distinctive regions. The area above the line is a one-phase region, while below this line, phase transitions can occur. For example, a mixture of overall composition xc will have a loop associated with it, like the one shown in figure 4.1, which just touches the critical line of the mixture at a unique point. The leg of the curve to the “left” of the critical point is referred to as the bubble line; while that to the right is termed the dew line. Phase equilibrium occurs between two phases at the point where the bubble line at one composition intersects the dew line; this requires two loops to be drawn of the sort shown in figure 4.1. A question naturally arises as to whether or not all binary systems exhibit continuous critical lines like that shown. In particular we are interested in the situation involving a nonvolatile solute dissolved in a supercritical fluid of high volatility.

On the number of interior peaks of solutions to a non-autonomous singularly perturbed Neumann problem

2009 ◽  
Vol 139 (2) ◽  
pp. 427-448 ◽  
Author(s):  
Chunyi Zhao
Keyword(s):  
Positive Integer ◽  
Neumann Problem ◽  
Smooth Function ◽  
Local Minimum ◽  
Singularly Perturbed ◽  
Minimum Point ◽  
Local Minimum Point ◽  
Image Position

We study the following non-autonomous singularly perturbed Neumann problem:where the index p is subcritical and a(x) is a positive smooth function in . We show that, given ε small enough, there exists a K(ε) such that, for any positive integer K ≤ K(ε), there always exists a solution with K interior peaks concentrating at a strict sth-order local minimum point of a.

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 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 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.

Algebraic Structures Associated to Orbifold Wreath Products

2010 ◽  
Vol 8 (2) ◽  
pp. 323-338
Author(s):  
Carla Farsi ◽  
Christopher Seaton
Keyword(s):  
Hopf Algebra ◽  
Wreath Product ◽  
Finite Groups ◽  
Lie Group ◽  
Wreath Products ◽  
Closed Manifold ◽  
Algebraic Structures ◽  
Present Structure ◽  
Smooth Closed Manifold ◽  
Connected Lie Group

AbstractWe present structure theorems in terms of inertial decompositions for the wreath product ring of an orbifold presented as the quotient of a smooth, closed manifold by a compact, connected Lie group acting almost freely. In particular we show that this ring admits λ-ring and Hopf algebra structures both abstractly and directly. This generalizes results known for global quotient orbifolds by finite groups.

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