Abstract:
Kortowski cheese (M nster type) was salted for 100, 75, 50 and 25% of their standard salting time, which is 48 hours. Cheese after 3 weeks of ripening and cheese immediately after salting were stored for 6 and 12 months at -27°C. Cheese of lower salting level ripened faster, both after salting and after frozen storage. The process of protein degradation occurred during frozen storage of ripe cheeses. The content of N-amino acid in ripe cheese after frozen storage and in cheese ripening after storage was almost twice as high as in the cheese that ripened after salting. Separations on Sephadex gel confirm the process of protein degradation during frozen storage of cheese. The conducted research indicated that frozen storage is recommended for Kortowski cheese of reduced salt content and the most favourable solution is to conduct the process of cheese ripening after thawing.

Abstract:
This paper contains results of studies on pollen morphology of 5 Polish species of the family Caprifoliaceae (genera Sambucus and Viburnum). The pollen has been examined with LM and SEM. Besides detailed descriptions, series of microphotographs are presented. All examined species have small to medium-sized grains, with prolate polar axis, reticulate ornamentation (at least in mesocolpia), relatively long ectoapertures and with often visible equatorial bridges. Extent of fusion of capita in muri allows to divide observed pollen grains into 2 subtypes: one for examined Sambucus species and for V. opulus, the second for V. lantana.

Abstract:
This paper presents further results of studies on pollen morphology of the family Caprifoliaceae. Besides detailed descriptions of 6 species of genera Linnaea and Lonicera, series of LM and SEM microphotographs are included. All examined species have pollen grains of medium- or large-sized, with oblate forms, echinate (microechinate) ornamentation and short ectoaperture, often margined by costa. Length of spines and density of their distribution on the tectum allow to distinguish Linnaea and Lonicera subtypes. The key for determination of 11 Caprifoliaceae species, based on pollen morphology is proposed too.

Abstract:
In this paper we consider dividend problem for an insurance company whose risk evolves as a spectrally negative L\'{e}vy process (in the absence of dividend payments) when Parisian delay is applied. The objective function is given by the cumulative discounted dividends received until the moment of ruin when so-called barrier strategy is applied. Additionally we will consider two possibilities of delay. In the first scenario ruin happens when the surplus process stays below zero longer than fixed amount of time $\zeta>0$. In the second case there is a time lag $d$ between decision of paying dividends and its implementation.

Abstract:
Consider two insurance companies (or two branches of the same company) that receive premiums at different rates and then split the amount they pay in fixed proportions for each claim (for simplicity we assume that they are equal). We model the occurrence of claims according to a Poisson process. The ruin is achieved when the corresponding two-dimensional risk process first leaves the positive quadrant. We will consider two scenarios of the controlled process: refraction and impulse control. In the first case the dividends are payed out when the two-dimensional risk process exits the fixed region. In the second scenario, whenever the process hits the horizontal line, it is reduced by paying dividends to some fixed point in the positive quadrant where it waits for the next claim to arrive. In both models we calculate the discounted cumulative dividend payments until the ruin. This paper is the first attempt to understand the effect of dependencies of two portfolios on the joint optimal strategy of paying dividends. For example in case of proportional reinsurance one can observe the interesting phenomenon that choice of the optimal barrier depends on the initial reserves. This is in contrast with the one-dimensional Cram\'{e}r-Lundberg model where the optimal choice of the barrier is uniform for all initial reserves.

Abstract:
We propose a new algorithm for computing validated bounds for the solutions to the first order variational equations associated to ODEs. The method uses a high order Taylor method as a predictor step and an implicit method based on the Hermite-Obreshkov interpolation as a corrector step. The proposed algorithm is an improvement of the $\mathcal C^1$-Lohner algorithm by Zgliczy\'nski and by its very construction it cannot produce worse bounds. As an application of the algorithm we give a computer assisted proof of the existence of an attractor in the Rossler system and we show that the attractor contains an invariant and uniformly hyperbolic subset on which the dynamics is chaotic, i.e. conjugated to the Bernoulli shift on two symbols.

Abstract:
In recent years there has been some focus on quasi-stationary behaviour of an one-dimensional L\'evy process, where we ask for the law $P(X_t\in dy | \tau^-_0>t)$ for $t\to\infty$ and $\tau_0^-=\inf\{t\geq 0: X_t<0\}$. In this paper we address the same question for so-called Parisian ruin time $\tau^\theta$, that happens when process stays below zero longer than independent exponential random variable with intensity $\theta$.

Abstract:
In this paper we consider the conformal type (parabolicity or non-parabolicity) of complete ends of revolution immersed in simply connected space forms of constant sectional curvature. We show that any complete end of revolution in the $3$-dimensional Euclidean space or in the $3$-dimensional sphere is parabolic. In the case of ends of revolution in the hyperbolic $3$-dimensional space, we find sufficient conditions to attain parabolicity for complete ends of revolution using their relative position to the complete flat surfaces of revolution.

Abstract:
In this paper we analyze so-called Parisian ruin probability that happens when surplus process stays below zero longer than fixed amount of time $\zeta>0$. We focus on general spectrally negative L\'{e}vy insurance risk process. For this class of processes we identify expression for ruin probability in terms of some other quantities that could be possibly calculated explicitly in many models. We find its Cram\'{e}r-type and convolution-equivalent asymptotics when reserves tends to infinity. Finally, we analyze few explicit examples.

Abstract:
A graph $G$ of order $n$ is called arbitrarily vertex decomposable if for each sequence $(n_1, ..., n_k)$ of positive integers such that $\sum _{i=1}^{k} n_i = n$, there exists a partition $(V_1, ..., V_k)$ of the vertex set of $G$ such that for every $i \in \{1, ...., k\}$ the set $V_i$ induces a connected subgraph of $G$ on $n_i$ vertices. A spider is a tree with one vertex of degree at least 3. We characterize two families of arbitrarily vertex decomposable spiders which are homeomorphic to stars with at most four hanging edges.