Volume 50 (2011): Issue 1 (December 2011)
Applied Mathematics and Informatics

This article deals with a possibility to identify parameters of a selected growth model of two populations coupled by a predator-prey interaction from a set of observed data. It starts with a brief description of the Gause-type model and of a property (interior equilibrium stability) important from a point of view of an application. Subsequently, data for four forms of the trophic function are simulated and then, a noise was added to the simulated data such that the coefficients of variation equal to 0.2, 0.3 and 0.4. For each data set, the parameters are estimated using a procedure implemented in the R-language package and the coordinates of equilibrium are computed. Then we can evaluate the effect of changing variation to the values of parameters and (un)stability of the equilibrium.

In this work we present the predator-prey model with Allee effect and Hawk and Dove tactics in fighting over caught prey implemented as fast strategy evolution dynamics. We extend the work of Auger, Parra, Morand and S´anchez (2002) using the prey population embodying Allee effect and analogously to this work we get two connected submodels with polymorphic and monomorphic predator population.We get much richer dynamics, in each submodel we find local bifurcations (saddle-node, supercritical Hopf caused by Allee effect and Bogdanov- -Takens) and a global bifurcation of limit cycles caused by the strategy evolution that is not possible in any of the submodels that can lead to a bluesky extinction of both populations.

One of the aims of the meta-analysis of clinical trials is to deter- mine the efficacy of a new type of treatment. This efficacy is commonly measured by the difference between the efficacy of a standard treatment and the new treatment. For binary data the difference can be measured by a probability difference. We investigate, by simulations and using box plots, the basic statistical properties of the point estimator of the probability difference of overall treatment effects in the meta-analysis based on multicentre trials for various chosen situations. This estimator was suggested in Dokoupilova (2011).

The k-nearest neighbour kernel density estimationmethod is a special type of the kernel density estimation method with the local choice of the bandwidth. An advantage of this estimator is that smoothing varies according to the number of observations in a particular region. The crucial problem is how to estimate the value of the parameter k. In the paper we discuss the problem of choosing the parameter k in a way that minimizes the value of the asymptotic mean integrated square error (AMISE). We define the class of the modified cosine densities that meet the requirements given by the AMISE. The results are compared in a simulation study.

The subject of this paper is linear and differential cryptanalysis of two rounds of the Advanced Encryption Standard (AES) with estimation of com- plexity for three-round AES attack. Presented linear attack is based on finding highly probable linear expressions and presented differential attack is based on finding specific bitwise differences. Data complexity of described linear and diffe- rential attack is 228 and 227, respectively, where 8 bits of subkey are recovered. Minimal complexity of linear attack on three-round AES is bigger than d × 260, where d is a small constant.

We continue in a direction of describing an algebraic structure of linear operators on infinite-dimensional complex Hilbert space ℋ. In [Paseka, J.- -Janda, J.: More on PT-symmetry in (generalized) effect algebras and partial groups, Acta Polytech. 51 (2011), 65-72] there is introduced the notion of a weakly ordered partial commutative group and showed that linear operators on H with restricted addition possess this structure. In our work, we are investigating the set of self-adjoint linear operators on H showing that with more restricted addition it also has the structure of a weakly ordered partial commutative group.

Inspired by the article of P. Grzegorzewski [The inclusion-exclusion principle for IF-events, Inform. Sci. 181 (2011), 536-546], who has worked two generalizations of the inclusion-exclusion principle for IF-events, a generalization of the inclusion-exclusion principle for mappings with values in semigroups is presented here. The main idea is in replacing the distributivity and idempotency laws, by one new axiom.