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1
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On the topological triviality along moduli of deformations of $J_{k,0}$ singularities

100%
Jaworski P.
Annales Polonici Mathematici
|
2000
|
tom 75
|
nr 3
193-212
EN
It is well known that versal deformations of nonsimple singularities depend on moduli. However they can be topologically trivial along some or all of them. The first step in the investigation of this phenomenon is to determine the versal discriminant (unstable locus), which roughly speaking is the obstacle to analytic triviality. The next one is to construct continuous liftable vector fields smooth far from the versal discriminant and to integrate them. In this paper we extend the results of J. Damon and A. Galligo, concerning the case of the Pham singularity ($J_{3,0}$ in Arnold's classification) (see [2, 3, 4]), and deal with deformations of general $J_{k,0}$ singularities.
2
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On the uniqueness of the quasihomogeneity

100%
Jaworski P.
Banach Center Publications
|
1999
|
tom 50
|
nr 1
163-167
EN
The aim of this paper is to show that the quasihomogeneity of a quasihomogeneous germ with an isolated singularity uniquely extends to the base of its analytic miniversal deformation.
3
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On the Witt rings of function fields of quasihomogeneous varieties

100%
Jaworski P.
Colloquium Mathematicum
|
1997
|
tom 73
|
nr 2
195-219
4
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On blowing up versal discriminants

100%
Jaworski P.
Banach Center Publications
|
1998
|
tom 44
|
nr 1
129-140
EN
It is well-known that the versal deformations of nonsimple singularities depend on moduli. The first step in deeper understanding of this phenomenon is to determine the versal discriminant, which roughly speaking is an obstacle for analytic triviality of an unfolding or deformation along the moduli. The goal of this paper is to describe the versal discriminant of $Z_{k,0}$ and $Q_{k,0}$ singularities basing on the fact that the deformations of these singularities may be obtained as blowing ups of certain deformations of $J_{k,0}$ singularities.
5
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Decompositions of hypersurface singularities oftype $J_{k,0}$

100%
Jaworski P.
Annales Polonici Mathematici
|
1994
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tom 59
|
nr 2
117-131
EN
Applications of singularity theory give rise to many questions concerning deformations of singularities. Unfortunately, satisfactory answers are known only for simple singularities and partially for unimodal ones. The aim of this paper is to give some insight into decompositions of multi-modal singularities with unimodal leading part. We investigate the $J_{k,0}$ singularities which have modality k - 1 but the quasihomogeneous part of their normal form only depends on one modulus.
6
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On uniform tail expansions of multivariate copulas and wide convergence of measures

100%
Jaworski P.
Applicationes Mathematicae
|
2006
|
tom 33
|
nr 2
159-184
EN
The theory of copulas provides a useful tool for modeling dependence in risk management. In insurance and finance, as well as in other applications, dependence of extreme events is particularly important, hence there is a need for a detailed study of the tail behaviour of multivariate copulas. We investigate the class of copulas having regular tails with a uniform expansion. We present several equivalent characterizations of uniform tail expansions. Next, basing on them, we determine the class of all possible leading parts of such expansions; we compute the leading parts of copulas popular in the literature, and discuss the statistical aspects of tail expansions.
7
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On uniform tail expansions of bivariate copulas

100%
Jaworski P.
Applicationes Mathematicae
|
2004
|
tom 31
|
nr 4
397-415
EN
The theory of copulas provides a useful tool for modelling dependence in risk management. The goal of this paper is to describe the tail behaviour of bivariate copulas and its role in modelling extreme events. We say that a bivariate copula has a uniform lower tail expansion if near the origin it can be approximated by a homogeneous function L(u,v) of degree 1; and it is said to have a uniform upper tail expansion if the associated survival copula has a lower tail expansion. In this paper we (1) introduce the notion of the uniform tail expansion of a bivariate copula; (2) describe the main properties of the leading part L(u,v) like two-monotonicity or concavity; (3) determine the set of all possible leading parts L(u,v); (4) compute the leading parts of the uniform tail expansions for the most popular copulas like gaussian, archimedean or BEV; (5) apply uniform tail expansions in estimating the extreme risk of a portfolio consisting of long positions in risky assets.
8
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On Truncation Invariant Copulas and their Estimation

100%
Jaworski P.
Dependence Modeling
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2017
|
tom 5
|
nr 1
133-144
EN
The paper deals with the family of irreducible left truncation invariant bivariate copulas, which admit a nontrivial lower tail dependence function. Such copulas, similarly as the Archimedean ones, are characterized by a functional parameter, a generator being an increasing convex function.We provide a nonparametric, piece-wise linear estimator of such generators.
9
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On Conditional Value at Risk (CoVaR) for tail-dependent copulas

100%
Jaworski P.
Dependence Modeling
|
2017
|
tom 5
|
nr 1
1-19
EN
The paper deals with Conditional Value at Risk (CoVaR) for copulas with nontrivial tail dependence. We show that both in the standard and the modified settings, the tail dependence function determines the limiting properties of CoVaR as the conditioning event becomes more extreme. The results are illustrated with examples using the extreme value, conic and truncation invariant families of bivariate tail-dependent copulas.
10
Content available remote

A geometric point of view on mean-variance models

100%
Jaworski P.
Applicationes Mathematicae
|
2003
|
tom 30
|
nr 2
217-241
EN
This paper deals with the mathematics of the Markowitz theory of portfolio management. Let E and V be two homogeneous functions defined on ℝⁿ, the first linear, the other positive definite quadratic. Furthermore let Δ be a simplex contained in ℝⁿ (the set of admissible portfolios), for example Δ : x₁+ ... + xₙ = 1, $x_i ≥ 0$. Our goal is to investigate the properties of the restricted mappings (V,E):Δ → ℝ² (the so called Markowitz mappings) and to classify them. We introduce the notion of a generic model (Δ,E,V) and investigate the equivalence of such models defined by continuous deformation.
11
Content available remote

Quadratic field extensions and residue homomorphisms of Witt rings

100%
Jaworski P.
Acta Arithmetica
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1993
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tom 64
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nr 4
303-323
12
Content available remote

On the versal discriminant of $J_{k,0}$ singularities

100%
Jaworski P.
Annales Polonici Mathematici
|
1996
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tom 63
|
nr 1
89-99
EN
It is well known that the versal deformations of nonsimple singularities depend on moduli. The first step in deeper understanding of this phenomenon is to determine the versal discriminant, which roughly speaking is an obstacle to analytic triviality of an unfolding or deformation along the moduli. The versal discriminant of the Pham singularity ($J_{3,0}$ in Arnold's classification) was thoroughly investigated by J. Damon and A. Galligo [2], [3], [4]. The goal of this paper is to continue their work and to describe the versal discriminant of a general $J_{k,0}$ singularity.
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