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On the Euler characteristic of the link of a weighted homogeneous mapping

100%
Dudziński P.
Annales Polonici Mathematici
|
2003
|
tom 81
|
nr 1
1-6
EN
The paper is concerned with an effective formula for the Euler characteristic of the link of a weighted homogeneous mapping $F:ℝⁿ → ℝ^k$ with an isolated singularity. The formula is based on Szafraniec's method for calculating the Euler characteristic of a real algebraic manifold (as the signature of an appropriate bilinear form). It is shown by examples that in the case of a weighted homogeneous mapping it is possible to make the computer calculations of the Euler characteristics much more effective.
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On the Euler characteristic of the real Milnor fibres of an analytic function

100%
Dudziński P.
Annales Polonici Mathematici
|
2003
|
tom 82
|
nr 1
45-50
EN
The paper is concerned with the relations between real and complex topological invariants of germs of real-analytic functions. We give a formula for the Euler characteristic of the real Milnor fibres of a real-analytic germ in terms of the Milnor numbers of appropriate functions.
3
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The choice of a lawyer as a special case of self-insurance-cum-protection

100%
Dudziński P.
Mathematica Applicanda
|
2013
|
tom 41
|
nr 2
PL
Analizujemy decyzję o wyborze prawnika jako szczególnym przypadku samoubezpieczenia z ochroną, gdy koszt prawnika zostanie spłacony w przypadku wygrania procesu. Problem został wprowadzony przez Sevi i Yafil(2005) w kontekście obrony, która wymaga założenia, że wielkość strat nie zależy od wysiłku (poziom wydatków na adwokata ).
EN
We consider decision about the choice of a lawyer as a particular case of self-insurance-cum-protection when the lawyer's cost is repaid  in case of victory.   The problem was introduced by Sevi and Yafil (2005) in the context of self-protection, which requires assumption that the size of  loss does not depend on effort (level of the expenditure on lawyer). In this paper we drop that assumption and our model includes possibility that both loss and  probability of  incuring a loss depend on effort.  We compare effort in our case with the standard one and prove that repayment is good incentive to invest more in modified SICP. We also show  that unlike the standard cases of SP and SICP, the level of effort is monotone in risk aversion. We prove that in our model DARA implies that lawyer's service is a normal good, which is intuitive. We show that for certain type of increase in risk aversion, the reimbursement effect is stronger then the risk aversion effect. For other changes in risk aversion there is a probability threshold that if the probability of a loss is below that level, then risk-aversion effect prevails. For higher initial probabilities, reimbursement effect is stronger.
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