The monoid consisting of Kuratowski operations

The paper fills gaps in knowledge about Kuratowski operations which are already in the literature. The Cayley table for these operations has been drawn up. Techniques, using only paper and pencil, to point out all semigroups and its isomorphic types are applied. Some results apply only to topology, one can not bring them out, using only properties of the complement and a closure-like operation. The arguments are by systematic study of possibilities.Comment: We are going to submit the article to a journa

Zastosowania teorii kategorii w topologii ogólnej

This dissertation is concerned with some notions of category theory in application to general topology and investigates to what extent the arguments making use of those notions can replace a more traditional approach. One of the generic examples is the notion of Fraisse sequence. Quite a number of categories consist of algebraic objects: groups, Abelian groups and monoids with group, Abelian group and monoid homomorphisms, respectively. An example of a monoid consisting of topological operations is the Kuratowski monoid. The dissertation is organized as follows. In subsection 1.2.1 we give some comments on ways of making use of concepts such as a tree, the space of branches, an ultra-metric and so forth. In section 2 we provide axioms for categories, basic definitions, notations and examples. We draw attention to a particular covariant functor, namely a diagram. At the beginning of section 3 we remind basic ideas concerning the limit of inverse sequence. Then we analyze a proof of Knaster-Reichbach theorem about extendibility of a mapping between two closed, nowhere dense subsets of the Cantor set and its version for scattered compact metric spaces. In section 4 we focus on a symmetric Cantorval. Beside the Cantor set and the sum of a finite family of closed intervals it is one of the possible forms the set of subsums of a convergent series with positive terms can take. In section 5 we remind the notions of the amalgamation and the inverse amalgamation property as well as their special cases - pushout and pullback, respectively. We discuss 1 these notions in two particular categories: S e t category whose objects are sets and whose arrows are functions and T op category whose objects are topological spaces and whose arrows are continuous mappings. In section 6 we present the main idea of the dissertation, namely Fraisse sequences: definition, criterion of existence and some properties. In subsection 6.2 we focus on the category F in s e top/A, i.e. a comma-category related to the opposite category of the category F in s e t . In section 7 we discuss the structure and basic facts concerning Fraisse limits in the context of inverse sequences. We show a way in which the properties of projective universality and projective homogeneity combine with the concept of generic object (for a given subcategory). Section 8 comprises comments th a t will be included in joint papers with the supervisors concerning, among others, a category of finite non-empty sets and its linear version. We claim here th a t the Cantor set admits a unique strictly positive probability measure th a t takes rationa values on clopen sets and satisfies certain homogeneity condition

On the center of distances

We introduce the notion of a center of distances of a metric space and use it in a generalization of the theorem by John von Neumann on permutations of two sequences with the same set of cluster points in a compact metric space. This notion is also used to study sets of subsums of some sequences of positive reals, as well for some impossibility proofs. We compute the center of distances of the Cantorval, which is the set of subsums of the sequence 34 ,12 ,316 ,18 ,…,34 n ,24 n ,… 34,12,316,18,…,34n,24n,… , and for other related subsets of the reals

Adhesion and Activation of Blood Platelets on Laser-Structured Surfaces of Biomedical Metal Alloys

The laser surface modification of metallic implants presents a promising alternative to other surface modification techniques. A total of four alloyed metallic biomaterials were used for this study: medical steel (AISI 316L), cobalt–chromium–molybdenum alloy (CoCrMo) and titanium alloys (Ti6Al4V and Ti6Al7Nb). Samples of metallic biomaterials after machining were subjected to polishing or laser modification in two different versions. The results of surface modification were documented using SEM imaging and roughness measurement. After modification, the samples were sterilized with dry hot air, then exposed to citrate blood, washed with PBS buffer, fixed with glutaraldehyde, sputtered with a layer of gold and imaged using SEM to enable the quantification of adhered, activated and aggregated platelets on the surface of biomaterial samples. The average total number, counted in the field of view, of adhered platelets on the surfaces of the four tested biomaterials, regardless of the type of modification, did not differ statistically significantly (66 ± 81, 67 ± 75, 61 ± 70 and 57 ± 61 for AISI 316L, CoCrMo, Ti6Al4V and Ti6Al7Nb, respectively) and the average number of platelet aggregates was statistically significantly higher (p < 0.01) on the surfaces of AISI 316L medical steel (42 ± 53) and of the CoCrMo alloy (42 ± 52) compared to the surfaces of the titanium alloys Ti6Al4V (33 ± 39) and Ti6Al7Nb (32 ± 37). Remaining blood after contact was used to assess spontaneous platelet activation and aggregation in whole blood by flow cytometry. An in-depth analysis conducted on the obtained results as a function of the type of modification indicates small but statistically significant differences in the interaction of platelets with the tested surfaces of metallic biomaterials

Comprehensive Biological Evaluation of Biomaterials Used in Spinal and Orthopedic Surgery

Biological acceptance is one of the most important aspects of a biomaterial and forms the basis for its clinical use. The aim of this study was a comprehensive biological evaluation (cytotoxicity test, bacterial colonization test, blood platelets adhesion test and transcriptome and proteome analysis of Saos-2 cells after contact with surface of the biomaterial) of biomaterials used in spinal and orthopedic surgery, namely, Ti6Al4V ELI (Extra Low Interstitials), its modified version obtained as a result of melting by electron beam technology (Ti6Al4V ELI-EBT), polyether ether ketone (PEEK) and polished medical steel American Iron and Steel Institute (AISI) 316L (the reference material). Biological tests were carried out using the osteoblasts-like cells (Saos-2, ATCC HTB-85) and bacteria Escherichia coli (DH5α). Results showed lack of cytotoxicity of all materials and the surfaces of both Ti6Al4V ELI and PEEK exhibit a significantly higher resistance to colonization with E. coli cells, while the more porous surface of the same titanium alloy produced by electron beam technology (EBT) is more susceptible to microbial colonization than the control surface of polished medical steel. None of the tested materials showed high toxicity in relation to E. coli cells. Susceptibility to platelet adhesion was very high for polished medical steel AISI 316L, whilst much lower for the other biomaterials and can be ranked from the lowest to the highest as follows: PEEK &lt; Ti6Al4V ELI &lt; Ti6Al4V ELI-EBT. The number of expressed genes in Saos-2 cells exposed to contact with the examined biomaterials reached 9463 genes in total (ranging from 8455 genes expressed in cells exposed to ELI to 9160 genes in cells exposed to PEEK). Whereas the number of differentially expressed proteins detected on two-dimensional electrophoresis gels in Saos-2 cells after contact with the examined biomaterials was 141 for PEEK, 223 for Ti6Al4V ELI and 133 for Ti6Al4V ELI-EBT. Finally, 14 proteins with altered expression were identified by mass spectrometry. In conclusion, none of the tested biomaterials showed unsatisfactory levels of cytotoxicity. The gene and protein expression analysis, that represents a completely new approach towards characterization of these biomaterials, showed that the polymer PEEK causes much more intense changes in gene and protein expression and thus influences cell metabolism