### About the Witt rings of function fields of algebroid quadratic quasihomogeneous surfaces.

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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.

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....

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...

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.

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...

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...

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, ${x}_{i}\ge 0$. 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...

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)...

Let V be a paracompact connected real analytic manifold of dimension 1 or 2, i.e. a smooth curve or surface. We consider it as a subset of some complex analytic manifold VC of the same dimension. Moreover by a prime divisor of V we shall mean the irreducible germ along V of a codimension one subvariety of VC which is an invariant of the complex conjugation. This notion is independent of the choice of the complexification VC. In the one-dimensional case prime divisors are just points, in the two-dimensional...

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.

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.

Performance of coherent reliability systems is strongly connected with distributions of order statistics of failure times of components. A crucial assumption here is that the distributions of possibly mutually dependent lifetimes of components are exchangeable and jointly absolutely continuous. Assuming absolute continuity of marginals, we focus on properties of respective copulas and characterize the marginal distribution functions of order statistics that may correspond to absolute continuous...

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