scholarly journals Equivalence of Milnor and Milnor-Lê fibrations for real analytic maps

2019 ◽  
Vol 30 (14) ◽  
pp. 1950078 ◽  
Author(s):  
José Luis Cisneros-Molina ◽  
Aurélio Menegon
Keyword(s):  
Vector Field ◽  
Fiber Bundle ◽  
General Setting ◽  
Critical Value ◽  
Complex Case ◽  
Linear Discriminant ◽  
Princeton University ◽  
Real Analytic ◽  
Analytic Singularities ◽  
The One

In [J. Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61 (Princeton University Press, Princeton, NJ, 1968).] Milnor proved that a real analytic map [Formula: see text], where [Formula: see text], with an isolated critical point at the origin has a fibration on the tube [Formula: see text]. Constructing a vector field such that (1) it is transverse to the spheres, and (2) it is transverse to the tubes, he “inflates” the tube to the sphere, to get a fibration [Formula: see text], but the projection is not necessarily given by [Formula: see text] as in the complex case. In the case [Formula: see text] has isolated critical value, in [J. L. Cisneros-Molina, J. Seade and J. Snoussi, Milnor fibrations and [Formula: see text]-regularity for real analytic singularities, Internat. J. Math. 21(4) (2010) 419–434.] it was proved that if the fibers inside a small tube are transverse to the sphere [Formula: see text], then it has a fibration on the tube. Also in [J. L. Cisneros-Molina, J. Seade and J. Snoussi, Milnor fibrations and [Formula: see text]-regularity for real analytic singularities, Internat. J. Math. 21(4) (2010) 419–434.], the concept of [Formula: see text]-regularity was defined, it turns out that [Formula: see text] is [Formula: see text]-regular if and only if the map [Formula: see text] is a fiber bundle equivalent to the one on the tube. In a more general setting, the corresponding facts are proved in [J. L. Cisneros-Molina, A. Menegon, J. Seade and J. Snoussi, Fibration theorems and [Formula: see text]-regularity for differentiable maps-germs with non-isolated critical value, Preprint (2017).], showing that if a locally surjective map [Formula: see text] has a linear discriminant [Formula: see text] with isolated singularity and a fibration on the tube [Formula: see text], then [Formula: see text] is [Formula: see text]-regular if and only if the map [Formula: see text] (with [Formula: see text] the radial projection of [Formula: see text] on [Formula: see text]) is a fiber bundle equivalent to the one on the tube. In this paper, we generalize this result for an arbitrary linear discriminant by constructing a vector field [Formula: see text] which inflates the tube to the sphere in a controlled way, it satisfies properties analogous to the vector field constructed by Milnor in the complex setting: besides satisfying (1) and (2) above, it also satisfies that [Formula: see text] is constant on the integral curves of [Formula: see text].

MILNOR FIBRATIONS AND d-REGULARITY FOR REAL ANALYTIC SINGULARITIES

2010 ◽  
Vol 21 (04) ◽  
pp. 419-434 ◽  
Author(s):  
J. L. CISNEROS-MOLINA ◽  
J. SEADE ◽  
J. SNOUSSI
Keyword(s):  
Fiber Bundle ◽  
Critical Value ◽  
Singular Set ◽  
Small Sphere ◽  
Milnor Fibration ◽  
Analytic Varieties ◽  
Real Analytic ◽  
Analytic Singularities ◽  
Analytic Maps ◽  
Do So

We study Milnor fibrations of real analytic maps [Formula: see text], n ≥ p, with an isolated critical value. We do so by looking at a pencil associated canonically to every such map, with axis V = f-1(0). The elements of this pencil are all analytic varieties with singular set contained in V. We introduce the concept of d-regularity, which means that away from the axis each element of the pencil is transverse to all sufficiently small spheres. We show that if V has dimension 0, or if f has the Thom af-property, then f is d-regular if and only if it has a Milnor fibration on every sufficiently small sphere, with projection map f/‖f‖. Our results include the case when f has an isolated critical point. Furthermore, we show that if f is d-regular, then its Milnor fibration on the sphere is equivalent to its fibration on a Milnor tube. To prove these fibration theorems we introduce the spherefication map, which is rather useful to study Milnor fibrations. It is defined away from V; one of its main properties is that it is a submersion if and only if f is d-regular. Here restricted to each sphere in ℝn the spherefication gives a fiber bundle equivalent to the Milnor fibration.

On topological invariants of real analytic singularities

2001 ◽  
Vol 130 (1) ◽  
pp. 13-24
Author(s):  
ZBIGNIEW SZAFRANIEC
Keyword(s):  
Analytic Functions ◽  
Closed Ball ◽  
The Other ◽  
Complex Case ◽  
Small Radius ◽  
Topological Invariants ◽  
Milnor Fibre ◽  
Real Analytic ◽  
Analytic Singularities ◽  
Regular Value

Let F = (f1, …, fm): (Kn, 0) → (Km, 0), where K is either R or C, be an analytic mapping defined in a neighbourhood of the origin. Let Br ⊂ Kn be a closed ball of small radius r centred at the origin. For any regular value y ∈ Km close to the origin, the fibre Wy = F−1(y) ∩ Br is called the Milnor fibre of F. We assume that m [les ] n, because in the other case Wy is void.Several authors investigated the topology of the Milnor fibres. Let us recall the most important results in the complex case. Let [Oscr ]C,0 denote the ring of germs of analytic functions f: (Cn, 0) → C.

Some Properties of Symplectic Manifolds S on the Projective Tangent Bundle PTM

2011 ◽  
Vol 187 ◽  
pp. 483-486
Author(s):  
Yong He ◽  
Xiao Ying Lu ◽  
Wei Na Lu
Keyword(s):  
Vector Field ◽  
Symplectic Manifold ◽  
Tangent Bundle ◽  
Fiber Bundle ◽  
Symplectic Manifolds ◽  
Lie Derivative ◽  
The Relationship

In this paper, we show the relationship between 2-form of the two projective tangent bundle and the relationship between 2-form on projective tangent bundle and 1-form on by using the theory of fiber bundle and the properties of symplectic manifold of the projective tangent bundle . Moreover, we derived a simpler formula of Lie derivative of a special vector field, which is on the projective tangent bundle.

Critical Value-Based Power Options Pricing Problems in Uncertain Financial Markets

2021 ◽  
pp. 2150002
Author(s):  
Guimin Yang ◽  
Yuanguo Zhu
Keyword(s):  
Financial Markets ◽  
Financial Market ◽  
Numerical Experiments ◽  
Critical Value ◽  
The Other ◽  
Options Pricing ◽  
Underlying Asset ◽  
Pricing Problems ◽  
Fractional Differential ◽  
The One

Compared with investing an ordinary options, investing the power options may possibly yield greater returns. On the one hand, the power option is the best choice for those who want to maximize the leverage of the underlying market movements. On the other hand, power options can also prevent the financial market changes caused by the sharp fluctuations of the underlying assets. In this paper, we investigate the power option pricing problem in which the price of the underlying asset follows the Ornstein–Uhlenbeck type of model involving an uncertain fractional differential equation. Based on critical value criterion, the pricing formulas of European power options are derived. Finally, some numerical experiments are performed to illustrate the results.

scholarly journals On implicit variables in optimization theory

2021 ◽  
Vol Volume 2 (Original research articles) ◽  
Author(s):  
Matúš Benko ◽  
Patrick Mehlitz
Keyword(s):  
Optimization Problems ◽  
Constraint Qualifications ◽  
Optimization Theory ◽  
Mathematical Optimization ◽  
General Setting ◽  
Necessary Optimality Conditions ◽  
Detailed Comparison ◽  
Mordukhovich Stationarity ◽  
Stationarity Conditions ◽  
The One

Implicit variables of a mathematical program are variables which do not need to be optimized but are used to model feasibility conditions. They frequently appear in several different problem classes of optimization theory comprising bilevel programming, evaluated multiobjective optimization, or nonlinear optimization problems with slack variables. In order to deal with implicit variables, they are often interpreted as explicit ones. Here, we first point out that this is a light-headed approach which induces artificial locally optimal solutions. Afterwards, we derive various Mordukhovich-stationarity-type necessary optimality conditions which correspond to treating the implicit variables as explicit ones on the one hand, or using them only implicitly to model the constraints on the other. A detailed comparison of the obtained stationarity conditions as well as the associated underlying constraint qualifications will be provided. Overall, we proceed in a fairly general setting relying on modern tools of variational analysis. Finally, we apply our findings to different well-known problem classes of mathematical optimization in order to visualize the obtained theory. Comment: 34 pages

Moderate deviations for linear eigenvalue statistics of β-ensembles

2021 ◽  
pp. 2250017
Author(s):  
Fuqing Gao ◽  
Jianyong Mu
Keyword(s):  
Moderate Deviation ◽  
Main Ingredient ◽  
Moderate Deviation Principle ◽  
Uniform Estimates ◽  
Linear Eigenvalue Statistics ◽  
Deviation Principle ◽  
Eigenvalue Statistics ◽  
Real Analytic ◽  
The One ◽  
Analytic Potential

We establish a moderate deviation principle for linear eigenvalue statistics of [Formula: see text]-ensembles in the one-cut regime with a real-analytic potential. The main ingredient is to obtain uniform estimates for the correlators of a family of perturbations of [Formula: see text]-ensembles using the loop equations.

Commentary

2013 ◽  
pp. 17-21 ◽  
Author(s):  
Patrick C. Friman ◽  
Stacy Shaw
Keyword(s):  
The Other ◽  
Complex Case ◽  
Parent Child Interaction Therapy ◽  
Oppositional Defiant ◽  
Child Interaction ◽  
Other Hand ◽  
Reactive Attachment ◽  
Parent Child Interaction ◽  
The One ◽  
Parent Child

There is much to admire in this report of an adaptation of parent-child interaction therapy (PCIT) used to treat a uniquely complex case involving comorbid reactive attachment and oppositional defiant disorders. One the one hand, the paper reflects the remarkable potency and flexibility of PCIT. On the other hand, it reflects the clinical acumen and interpersonal dexterity of the clinicians who reported the case. We will discuss both of these aspects below....

scholarly journals On the Construction of Large Algebras Not Contained in the Image of the Borel Map

2019 ◽  
Vol 75 (1) ◽  
Author(s):  
Céline Esser ◽  
Gerhard Schindl
Keyword(s):  
Sequence Space ◽  
Analytic Functions ◽  
General Setting ◽  
Smooth Functions ◽  
The Real ◽  
Cauchy Product ◽  
Real Analytic Functions ◽  
Borel Map ◽  
Real Analytic ◽  
The Stability

AbstractThe Borel map $$j^{\infty }$$j∞ takes germs at 0 of smooth functions to the sequence of iterated partial derivatives at 0. It is well known that the restriction of $$j^{\infty }$$j∞ to the germs of quasianalytic ultradifferentiable classes which are strictly containing the real analytic functions can never be onto the corresponding sequence space. In a recent paper the authors have studied the size of the image of $$j^{\infty }$$j∞ by using different approaches and worked in the general setting of quasianalytic ultradifferentiable classes defined by weight matrices. The aim of this paper is to show that the image of $$j^{\infty }$$j∞ is also small with respect to the notion of algebrability and we treat both the Cauchy product (convolution) and the pointwise product. In particular, a deep study of the stability of the considered spaces under the pointwise product is developed.

Path-integral approach to anomalies at finite temperature

1990 ◽  
Vol 68 (1) ◽  
pp. 96-103 ◽  
Author(s):  
T. F. Treml
Keyword(s):  
Vector Field ◽  
Path Integral ◽  
Finite Temperature ◽  
Dimensional Regularization ◽  
Flat Space ◽  
Integral Approach ◽  
Chiral Anomaly ◽  
Conformal Anomaly ◽  
Path Integral Approach ◽  
The One

The non-Abelian chiral anomaly for a fermion interacting with an external vector field in any even dimension and the conformal anomaly, in the limit of flat space–time, for a self-interacting scalar field are shown to be independent of temperature using a simple path-integral approach that employs dimensional regularization. The chiral anomaly is used as an example to show that the methods used to study the dimensionally regularized anomaly at finite temperature are readily transferable to the case of ζ-function regularization. The conformal anomaly in (super) string theory at finite temperature is briefly discussed in the light of known results. Some subtleties concerning the use of infrared cutoffs in a dimensionally regularized approach to the computation of the one-loop effective action at finite temperature are considered in an appendix.

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