A mathematical procedure enabling the transformation of an arbitrary tessellation of a surface into a bi-colored, complete graph is introduced. Polygons constituting the tessellation are represented by vertices of the graphs. Vertices of the graphs are connected by two kinds of links/edges, namely, by a green link, when polygons have the same number of sides, and by a red link, when the polygons have a different number of sides. This procedure gives rise to a semi-transitive, complete, bi-colored Ramsey graph. The Ramsey semi-transitive number was established as $R_{trans}\left(3,3\right)=5$ Shannon entropies of the tessellation and graphs are introduced. Ramsey graphs emerging from random Voronoi and Poisson Line tessellations were investigated. The limits $\zeta =\underset{N\to \infty }{\lim }{\displaystyle \frac{N_g}{N_r}}$, where N is the total number of green and red seeds, $N_g$ and $N_r$, were found $\zeta =$ 0.272 ± 0.001 (Voronoi) and $\zeta =$ 0.47 ± 0.02 (Poisson Line). The Shannon Entropy for the random Voronoi tessellation was calculated as $S=$ 1.690 ± 0.001 and for the Poisson line tessellation as S = 1.265 ± 0.015. The main contribution of the paper is the calculation of the Shannon entropy of the random point process and the establishment of the new bi-colored Ramsey graph on top of the tessellations.

A mathematical procedure enabling the transformation of an arbitrary tessellation of a surface into a bi-colored complete graph is introduced. Polygons constituting the tessellation are represented by vertices of the graphs. Vertices of the graphs are connected by two kinds of links/edges, namely, by a green link, when polygons have the same number of sides, and by a red link, when the polygons have a different number of sides. This procedure gives rise to a semi-transitive, complete, bi-colored Ramsey graph. The Ramsey number was established as R_trans (3,3)=5. Shannon entropies of the tessellation and graphs are introduced. Ramsey graphs emerging from random Voronoi and Poisson Line tessellations were investigated. The limits ζ=lim┬(N→∞)⁡〖N_g/N_r 〗, where N is the total number of green and red seeds, N_g and N_r, were found ζ=0.272±0.001 (Voronoi) and ζ= 0.47±0.02 (Poisson Line). The Shannon Entropy for the random Voronoi tessellation was calculated as S= 1.690±0.001 and for the Poisson line tessellation as S =1.265±0.015.

We investigated the time evolution of the symmetry of symbols constituting alphabets rooted in the Phoenician script. A diversity of quantitative measures of symmetry of graphemes appearing in Phoenician, Western Greek, Etruscan from Marsiliana, Archaic Etruscan, Neo-Etruscan, Euclidian Greek, Archaic and Classical Latin and Proto-Hebrew scripts, constituting the Phoenician script family, were calculated. The same measures were established for the Hebrew/Ashurit and English scripts. The Shannon-like measures of symmetry were computed. The Shannon diversity index was calculated. Our findings indicate that the Shannon diversity index increased with time in a monotonic way for the studied scripts. The diversity of symmetry groups inherent for addressed alphabets grows with time. We also introduced the symmetry factor of the alphabet. The symmetry factor quantifies the averaged level of symmetrization of the alphabet and the possible parsimony of graphical information necessary for the drawing of the entire set of graphemes constituting the alphabet. We found that the symmetry factor is decreased with time for the alphabets rooted in the Phoenician script. This means that the average level of symmetrization of the studied alphabet increases with time. The parsimony of graphical information necessary for writing graphemes is consequently increased with time. The values of the symmetry factor calculated for the addressed scripts are close to one another, with the pronounced exception of the Hebrew/Ashurit script. Our study supplies the arguments for the point of view, according to which the modern Hebrew/Ashurit script did not emerge from the Phoenician one.

Extruded polypropylene (PP) films were exposed to cold air plasma treatment, which resulted in significant changes in their bulk properties. The maximal elongation, ultimate tensile strength (UTS), and toughness of the films were increased. The toughness of the films increased from $U_{T0}=(3323\pm 400) \mathrm{M}\mathrm{P}\mathrm{a}$ to $U_{T\_PT}=(4434\pm 400) \mathrm{M}\mathrm{P}\mathrm{a}$, which is due to the growth of both the maximal elongation and the UTS of the plasma-treated samples. We relate the improvement of the mechanical properties of PP to the morphological transformations revealed in the plasma-treated PP films. Plasma treatment of PP samples was also followed by the modification of their surface properties. Plasma treatment resulted in hydrophilization of PP films followed by hydrophobic recovery. The bulk and surface properties of the plasma-treated PP films evolve with time. The following hierarchy of the temporal scales related to the studied relaxation processes is established: ${\tau }_{HR}>{\tau }_{\epsilon }={\tau }_T={\tau }_{UTS}>{\tau }_E$, where ${\tau }_{HR}, {\tau }_{\epsilon }, {\tau }_T, {\tau }_{UTS}$ and ${\tau }_E$ are the time scales of the change in the apparent contact angle (hydrophobic recovery), elongation, toughness, ultimate tensile strength, and Young modulus, respectively. The longest of the relaxation times is related to the surface processes, i.e., hydrophobic recovery. The stress–strain curves of the untreated virgin and plasma-treated PP are well described with the twin-slope linear dependencies. The post-plasma-treatment recovery of the tangent modulus is reported. Cold plasma treatment of polypropylene produces surface oxidation and functionalization, evidenced by the emergence of C–O, C=O, and COOH functionalities.

The paper addresses the phenomena of negative effective mass and negative effective density emerging in systems driven by entropic elastic forces. The elasticity of polymers is, at least partially, of entropic origin, and it represents the tendency of a polymer to evolve into a more probable state, rather than into one of lower potential energy. Entropy forces are temperature-dependent; thus, the temperature dependence of the effective mass and effective density arises. The effect of the negative effective mass is a resonance effect, emerging in core–shell mechanical systems, which takes place when the frequency of the harmonic external force acting on a core–shell system connected by an ideal spring approaches from above to the eigen-frequency of the system. We address the situation when the ideal spring connecting the core to the shell is made from a polymer material, and its elasticity is of an entropic origin. The effective mass is calculated, and it is temperature-dependent. The chain of core–shell units connected with a polymer spring is studied. The effective density of the spring is temperature-dependent. Optical and acoustical branches of vibrations are elucidated. The negative mass and density become attainable under the variation of the temperature of the system. In the situation when only one of the springs demonstrates temperature dependence, entropic behavior is investigated. Exemplifications of the effect are addressed.

Plane tessellations naturally generate local combinatorial invariants such as vertex degree and vertex type, which encode essential geometric and topological information. In the present work, these invariants are used to construct complete bicolored graphs whose edge coloring is induced intrinsically by the equality or inequality of local tessellation properties. Two classes of structured Ramsey graphs are introduced: vertex-degree graphs and vertex-type graphs. Unlike classical Ramsey colorings, the proposed colorings are not arbitrary but arise from equivalence relations generated by local geometric invariants. Consequently, the resulting graphs possess a partition-generated or semi-transitive structure in which one color forms disjoint complete cliques corresponding to equivalence classes, while the complementary color connects distinct classes. Within this framework, a sharp Ramsey-type theorem is established: every such graph on five vertices necessarily contains a monochromatic triangle. The combinatorial core of the theorem reduces to the statement that among five values either three coincide or three are pairwise distinct. Turán numbers for the introduced structured colorings are calculated and shown to possess a characteristic asymmetry that is absent in classical extremal graph theory. The relation-ship between geometric symmetries of tessellations and automorphism groups of the associated Ramsey graphs is analyzed, demonstrating that tessellation symmetries induce graph automorphisms, while additional purely combinatorial symmetries may emerge from degree degeneracies. The results establish a direct connection between tessellation geometry, equivalence relations, Ramsey theory, and extremal graph theory.