Fuzzy Number Addition with the Application of Horizontal Membership Functions.

The paper presents addition of fuzzy numbers realised with the application of the multidimensional RDM arithmetic and horizontal membership functions (MFs). Fuzzy arithmetic (FA) is a very difficult task because operations should be performed here on multidimensional information granules. Instead, a lot of FA methods use α-cuts in connection with 1-dimensional classical interval arithmetic that operates not on multidimensional granules but on 1-dimensional intervals. Such approach causes difficulties in calculations and is a reason for arithmetical paradoxes. The multidimensional approach allows for removing drawbacks and weaknesses of FA. It is possible thanks to the application of horizontal membership functions which considerably facilitate calculations because now uncertain values can be inserted directly into equations without using the extension principle. The paper shows how the addition operation can be realised on independent fuzzy numbers and on partly or fully dependent fuzzy numbers with taking into account the order relation and how to solve equations, which can be a difficult task for 1-dimensional FAs.

Genetic contribution to all cancers: the first demonstration using the model of breast cancers from Poland stratified by age at diagnosis and tumour pathology.

The aim of the study is to verify the hypothesis that genetic polymorphisms are associated with the predisposition to all malignancies. Using as a model breast cancers from the homogenous Polish population (West Pomeranian region) after stratification of 977 patients by age at diagnosis (under 51 years and above 50 years) and by tumour pathology (ductal cancers--low and high grade, lobular cancers, ER-positive/negative) we tested this hypothesis. Altogether 20 different groups of breast cancer cases have been analyzed. The results were compared to a group of unaffected controls that were matched by age, sex, ethnicity and geographical location and originated from families without cancers of any site among relatives. Molecular alterations selected for analyses included those which have been previously recognized as being associated with breast cancer predisposition. Statistically significant differences between the breast cancer cases and controls were observed in 19 of the 20 analyzed groups. Genetic changes were present in more than 90% of the breast cancer patients in 18 of 20 groups. The highest proportion of cases with constitutional changes-99.3% (139/140) was observed for lobular cancers. The number and type of genetic marker and/or the level of their association with the specific cancer predisposition was different between groups. Markers associated with majority of groups included: BRCA1, CHEK2, p53, TNRnTT, FGFRnAA, XPD CC/AA and XPD GG. Some markers appeared to be group specific and included polymorphisms in CDKN2A, CYP1B1, M3K nAA, and RS67.

Breast cancer susceptibility genes.

In 1999 it has been recognized that 3 BRCA1 abnormalities - 5382insC, C61G and 4153delA - constitute almost 90% of all germline mutations of this gene in Poland. Due to the above findings we started performing the cheap and quick large scale testing for BRCA1 mutations and, these days, we have almost 4,000 carriers diagnosed and under direct or indirect supervision what is probably the largest number in the world. Additionally, the above results pushed us to hypothesize that genetic homogeneity will be seen in Poland in studies of other genes. Actually, the next studies allowed us to identify genes / changes associated with moderate / low breast cancer risk and showed, similarly to BRCA1, high level of genetic homogeneity. This series included BRCA2, C5972T, CHEK2 del5395; 1100delC, I157T or IVS2 + 1G > A, CDKN2A (p16) A148T, XPD Asp312Asn and Lys751Gln, CYP1B1 R48G, A119S and L43V. The results of the above studies led us in 2004 already to hypothesize that >90% of all cancers have genetic (constitutional) background. Two years later we were able to show a panel of markers covering 92% of consecutive breast cancers in Poland, and we formulated the hypothesis that all cancers have a genetic background. These days we are demonstrating for the first time that genetic components to malignancy play a role in all cancers. We are presenting it on examples of late-onset breast cancers from Poland, but it seems to be justified to expect that similar results can be achieved from other malignancies.