scholarly journals Stratified critical points on the real Milnor fibre and integral-geometric formulas

2015 ◽  
Vol 13 ◽  
Author(s):  
Nicolas Dutertre
Keyword(s):  
Critical Points ◽  
The Real ◽  
Milnor Fibre

scholarly journals Analysis of the atmosphere behaviour in the proximities of an orographic obstacle

1995 ◽  
Vol 2 (1) ◽  
pp. 30-48 ◽  
Author(s):  
E. Hernández ◽  
J. Díaz ◽  
L. C. Cana ◽  
A. García
Keyword(s):  
Boundary Layer ◽  
Numerical Simulations ◽  
Critical Points ◽  
Complex Terrain ◽  
Laboratory Experiments ◽  
Real Experiment ◽  
The Real ◽  
Linear And Nonlinear ◽  
Atmospheric Variables

Abstract. The atmospheric behaviour near an orographic obstacle has been thoroughly studied in the last decades. The first papers in this field were mainly theoretical, being more recent the laboratory experiments which represented that behaviour in ideal conditions. The numerical simulations have been addressed lately thanks to the development of computers. But the study of meteorology in complex terrain has lacked experiments in the atmosphere to understand the real influence the relief has on it. In this paper the problem has been considered from the last perspective, and so, seasons of measure of the atmospheric variables within the boundary layer have been organized with the goal of checking existing theories and bringing right conclusions from real experiment in the atmosphere. Controverted aspects of linear and nonlinear theories, as the location of critical points upwind and downwind of an orographic obstacle, will be analyzed. The results obtained show a large adequacy between the forecasted behaviour and the experimentally detected.

The double cusp has five minima

1978 ◽  
Vol 84 (3) ◽  
pp. 537-538 ◽  
Author(s):  
J. Callahan
Keyword(s):  
Critical Points ◽  
Standard Form ◽  
Singularity Theory ◽  
The Other ◽  
Universal Unfolding ◽  
The Real ◽  
Level Curves ◽  
Intersection Points ◽  
The One ◽  
Image Position

The double cusp is the real, compact, unimodal singularitysee (2), (4). Functions in a universal unfolding of the double cusp can have nine non-degenerate critical points near the origin, but no more. Index considerations show that precisely four of the nine are saddles, and it has long been part of the folklore of singularity theory that one of the other five must be a maximum. Indeed, a standard form of the unfolded double cusp (1), (3) is a function having a pair of intersecting ellipses as one of its level curves; see Fig. 1(a). There are saddles at the four intersection points, a maximum inside the central quadrilateral, and a minimum inside each of the other four finite regions bounded by the ellipses. The rest of Fig. 1 suggests, however, that a deformation of this function (in which one of the saddles drops below the level of the other three) might turn the maximum into a fifth minimum. The following proposition shows that a function similar to the one in Fig. 1(d) can be realized in an unfolding of the double cusp.

Hyperplane Singularities of Analytic Functions of Several Complex Variables

1997 ◽  
Vol 4 (2) ◽  
pp. 163-184
Author(s):  
M. Shubladze
Keyword(s):  
Critical Points ◽  
Analytic Functions ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Algebraic Expression ◽  
Isolated Singularities ◽  
Milnor Fibre ◽  
New Class

Abstract A new class of non-isolated singularities called hyperplane singularities is introduced. Special deformations with simplest critical points are constructed and an algebraic expression for the number of Morse points is given. The topology of the Milnor fibre is completely studied.

ANALYTIC HYPOELLIPTICITY FOR SUMS OF SQUARES IN THE PRESENCE OF SYMPLECTIC NON TREVES STRATA

2019 ◽  
Vol 19 (6) ◽  
pp. 1877-1888 ◽  
Author(s):  
Antonio Bove ◽  
Marco Mughetti
Keyword(s):  
Critical Points ◽  
Sums Of Squares ◽  
Characteristic Variety ◽  
The Real ◽  
Real Analytic ◽  
Funct Anal ◽  
Analytic Regularity ◽  
Analytic Hypoellipticity

In Albano, Bove and Mughetti [J. Funct. Anal. 274(10) (2018), 2725–2753]; Bove and Mughetti [Anal. PDE 10(7) (2017), 1613–1635] it was shown that Treves conjecture for the real analytic hypoellipticity of sums of squares operators does not hold. Models were proposed where the critical points causing a non-analytic regularity might be interpreted as strata. We stress that up to now there is no notion of stratum which could replace the original Treves stratum. In the proposed models such ‘strata’ were non-symplectic analytic submanifolds of the characteristic variety. In this note we modify one of those models in such a way that the critical points are a symplectic submanifold of the characteristic variety while still not being a Treves stratum. We show that the operator is analytic hypoelliptic.

Geometrical conditions for the stability of orbits in planar systems

1996 ◽  
Vol 120 (3) ◽  
pp. 499-519 ◽  
Author(s):  
R. A. Garcia ◽  
A. Gasull ◽  
A. Guillamon
Keyword(s):  
Vector Field ◽  
Periodic Orbit ◽  
Critical Points ◽  
Real Plane ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
The Real ◽  
Planar Systems ◽  
The Stability ◽  
Derivatives Of

AbstractGiven a vector field X on the real plane, we study the influence of the curvature of the orbits of ẋ = X┴(x) in the stability of those of the system x˙ = X(x). We pay special attention to the case in which this curvature is negative in the whole plane. Under this assumption, we classify the possible critical points and give a criterion for a point to be globally asymptotically stable. In the general case, we also provide expressions for the first three derivatives of the Poincaré map associated to a periodic orbit in terms of geometrical quantities.

scholarly journals The spectrum of the Hamiltonian with a PT-symmetric periodic optical potential

2017 ◽  
Vol 15 (01) ◽  
pp. 1850008 ◽  
Author(s):  
O. A. Veliev
Keyword(s):  
Critical Point ◽  
Critical Points ◽  
Optical Potential ◽  
Mathematical Proof ◽  
Hill Operator ◽  
The Real ◽  
The Hill ◽  
Degeneration Point

We give a complete description, provided with a mathematical proof, of the shape of the spectrum of the Hill operator with potential [Formula: see text], where [Formula: see text] We prove that the second critical point [Formula: see text], after which the real parts of the first and second bands disappear, is a number between [Formula: see text] and [Formula: see text]. Moreover, we prove that [Formula: see text] is the degeneration point for the first periodic eigenvalue. Besides, we give a scheme by which one can find arbitrary precise value of the second critical point as well as the [Formula: see text]th critical points after which the real parts of the [Formula: see text]th and [Formula: see text]th bands disappear, where [Formula: see text]

scholarly journals Gorenstein-duality for one-dimensional almost complete intersections – with an application to non-isolated real singularities

2014 ◽  
Vol 158 (2) ◽  
pp. 249-268 ◽  
Author(s):  
DUCO VAN STRATEN ◽  
THORSTEN WARMT
Keyword(s):  
Complete Intersection ◽  
Critical Points ◽  
Real Hypersurface ◽  
Complete Intersections ◽  
Hypersurface Singularity ◽  
One Dimensional ◽  
Milnor Fibre ◽  
Real Curves ◽  
Real Singularities ◽  
Special Case

AbstractWe give a generalisation of the duality of a zero-dimensional complete intersection for the case of one-dimensional almost complete intersections, which results in a Gorenstein module M = I/J. In the real case the resulting pairing has a signature, which we show to be constant under flat deformations. In the special case of a non-isolated real hypersurface singularity f, with a one-dimensional critical locus, we relate the signature on the Jacobian module I/Jf to the Euler characteristic of the positive and negative Milnor fibre, generalising the result for isolated critical points. An application to real curves in ℙ2(ℝ) of even degree is given.

The relationship of the scleractinian corals to the rugose corals

Paleobiology ◽  
1980 ◽  
Vol 6 (02) ◽  
pp. 146-160 ◽  
Author(s):  
William A. Oliver
Keyword(s):  
Critical Points ◽  
Scleractinian Corals ◽  
Convincing Evidence ◽  
Early Triassic ◽  
Rugose Corals ◽  
History Of ◽  
The Subject ◽  
Relationship Of ◽  
The Relationship ◽  
Biologic Factors

The Mesozoic-Cenozoic coral Order Scleractinia has been suggested to have originated or evolved (1) by direct descent from the Paleozoic Order Rugosa or (2) by the development of a skeleton in members of one of the anemone groups that probably have existed throughout Phanerozoic time. In spite of much work on the subject, advocates of the direct descent hypothesis have failed to find convincing evidence of this relationship. Critical points are:(1) Rugosan septal insertion is serial; Scleractinian insertion is cyclic; no intermediate stages have been demonstrated. Apparent intermediates are Scleractinia having bilateral cyclic insertion or teratological Rugosa.(2) There is convincing evidence that the skeletons of many Rugosa were calcitic and none are known to be or to have been aragonitic. In contrast, the skeletons of all living Scleractinia are aragonitic and there is evidence that fossil Scleractinia were aragonitic also. The mineralogic difference is almost certainly due to intrinsic biologic factors.(3) No early Triassic corals of either group are known. This fact is not compelling (by itself) but is important in connection with points 1 and 2, because, given direct descent, both changes took place during this only stage in the history of the two groups in which there are no known corals.

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