Packing triangles in low degree graphs and indifference graphs

We consider the problems of finding the maximum number of vertex-disjoint triangles (VTP) and edge-disjoint triangles (ETP) in a simple graph. Both problems are NP-hard. The algorithm with the best approximation guarantee known so far for these problems has ratio $3/2 + ɛ$, a result that follows from a more general algorithm for set packing obtained by Hurkens and Schrijver in 1989. We present improvements on the approximation ratio for restricted cases of VTP and ETP that are known to be APX-hard: we give an approximation algorithm for VTP on graphs with maximum degree 4 with ratio slightly less than 1.2, and for ETP on graphs with maximum degree 5 with ratio 4/3. We also present an exact linear-time algorithm for VTP on the class of indifference graphs

Association between alcohol consumption and dietary supplement intake of students from the University of Sarajevo

Introduction: Dietary supplements are an important source of vitamins and minerals that may help prevent several disease-causing biological pathways involved in one-carbon metabolism, including the suppression of cell proliferation, oxidative stress, nitric oxide synthesis, and angiogenesis. The present study aimed to assess the association between the intake of folate, Vitamin B6, Vitamin B12, and minerals and the consumption of alcohol among university students. Methods: This study was participated by students aged 19-22 years from the University of Sarajevo between 2017 and 2018. Using a questionnaire, we interviewed in a week them to collect information regarding age, socio-economic status, alcohol consumption, and dietary supplement intake. Then, we investigated the association between the baseline intake of folate, B vitamins, and minerals and that of alcohol consumption. Results: Most students consumed Vitamin B supplements (32%) and folic acid (10%). Dietary multivitamins and minerals were less prevalent in more than a year, accounting for 186 (23.9%) and 174 (24.3%) students, respectively, than those in less than a year. In a year, <20% of students consumed multivitamins 129 (16.6%) and minerals 116 (15.3%). Meanwhile, 256 (27.1%) students consumed alcohol. The Chi-square test of independence showed that drinking habits and the intake of such dietary supplements had no association (p > 0.05). Conclusion: An extremely low percentage of the participating students in Canton Sarajevo used dietary supplements of Vitamin B, folate, multivitamins, and minerals. Moreover, alcohol consumption and dietary supplement intake were not associated. Further research is needed to establish the best cost-effective public health system to achieve a sufficient intake of dietary supplements

Packing triangles in low degree graphs and indifference graphs

We consider the problems of finding the maximum number of vertex-disjoint triangles (VTP) and edge-disjoint triangles (ETP) in a simple graph. Both problems are NP-hard. The algorithm with the best approximation guarantee known so far for these problems has ratio 3/2 + ε, a result that follows from a more general algorithm for set packing obtained by Hurkens and Schrijver in 1989. We present improvements on the approximation ratio for restricted cases of VTP and ETP that are known to be APX-hard: we give an approximation algorithm for VTP on graphs with maximum degree 4 with ratio slightly less than 1.2, and for ETP on graphs with maximum degree 5 with ratio 4/3. We also present an exact linear-time algorithm for VTP on the class of indifference graphs

Packing triangles in low degree graphs and indifference graphs

We consider the problems of finding the maximum number of vertex-disjoint triangles (VTP) and edge-disjoint triangles (ETP) in a simple graph. Both problems are NP-hard. The algorithm with the best approximation guarantee known so far for these problems has ratio $3/2 + ɛ$, a result that follows from a more general algorithm for set packing obtained by Hurkens and Schrijver in 1989. We present improvements on the approximation ratio for restricted cases of VTP and ETP that are known to be APX-hard: we give an approximation algorithm for VTP on graphs with maximum degree 4 with ratio slightly less than 1.2, and for ETP on graphs with maximum degree 5 with ratio 4/3. We also present an exact linear-time algorithm for VTP on the class of indifference graphs

Thermal Buckling and Free Vibration Analysis of Functionally Graded Plate Resting on an Elastic Foundation According to High Order Shear Deformation Theory Based on New Shape Function

Functionally graded square and rectangular plates of different thicknesses placed on the elastic foundation modeled according to the Winkler-Pasternak theory have been studied. The thermal and mechanical characteristics, apart from Poisson’s ratio, are considered to continuously differ through the thickness of the studied material as stated in a power-law distribution. A mathematical model of functionally graded plate which include interaction with elastic foundation is defined. The equilibrium and stability equations are derived using high order shear deformation theory that comprises various kinds of shape function and the von Karman nonlinearity. A new analytically integrable shape function has been introduced. Hamilton’s principle has been applied with the purpose of acquiring the equations of motion. An analytical method for identifying both natural frequencies and critical buckling temperature for cases of linear and nonlinear temperature change through the plate thickness has been established. In order to verify the derived theoretical results on numerical examples, an original program code has been implemented within software MATLAB. Critical buckling temperature and natural frequencies findings are shown below. Previous scientific research and papers confirms that presented both the theoretical formulation and the numerical results are accurate. The comparison has been made between newly established findings based on introduced shape function and the old findings that include 13 different shape functions available in previously published articles. The final part of the research provides analysis and conclusions related to the impact of the power-law index, foundation stiffness, and temperature gradient on critical buckling temperature and natural frequencies of the functionally graded plates

Cyanobacterial and algal abundance and biomass in cave biofilms and relation to environmental and biofilm parameters

Due to life in extreme environments, cyanobacteria and algae from cave biofilms that form at the entrances or deep inside the cave around artificial lights are of increasing interest to many scientists. It is well-known that many phototrophic microorganisms are first to colonize exposed substrata and produce the organic matter on which other biofilm constituents relly. Many studies dealing with phototrophic microorganisms from biofilms focus on the diversity and community composition of cyanobacteria and algae, while quantitative assessments are rarely implemented. Biofilm sampling was conducted in Degurić and Vernjikica Cave located in Western and Eastern Serbia, respectively. Ecological parameters (temperature, relative humidity, light intensity) and distance from the entrance were measured. Additionally, chlorophyll content, as well as biofilm parameters (water content, organic and inorganic matter) were determined. The abundance of phototrophic microorganisms was assessed on microscope slides which contained 1 mg of biofilm that was dehydrated for a short period of time and homogenized prior to slide preparation, and then rehydratated. The biomass of recorded cyanobacterial and algal taxa was calculated by applying geometric approximations and standard mathematical formulas. In Degurić Cave, at the sampling site where the highest biomass was documented, the higher diversity, water content and chlorophyll values were also recorded, while in Vernjikica Cave a high content of organic matter was documented. According to the multivariate analyses performed, the biomass of simple trichal Cyanobacteria, Bacillariophyta, and Xanthophyta was positively correlated with the content of organic matter in biofilm and light intensity, while coccoid and heterocytous Cyanobacteria and Chlorophyta showed a positive correlation with water content in the biofilm, relative humidity and distance from the entrance. The total biomass was positively correlated with the chlorophyll content, organic matter and light intensity, and negatively with the distance from the entrance

Cyanobacterial and algal abundance and biomass in cave biofilms and relation to environmental and biofilm parameters

Due to life in extreme environments, cyanobacteria and algae from cave biofilms that form at the entrances or deep inside the cave around artificial lights are of increasing interest to many scientists. It is well-known that many phototrophic microorganisms are first to colonize exposed substrata and produce the organic matter on which other biofilm constituents relly. Many studies dealing with phototrophic microorganisms from biofilms focus on the diversity and community composition of cyanobacteria and algae, while quantitative assessments are rarely implemented. Biofilm sampling was conducted in Degurić and Vernjikica Cave located in Western and Eastern Serbia, respectively. Ecological parameters (temperature, relative humidity, light intensity) and distance from the entrance were measured. Additionally, chlorophyll content, as well as biofilm parameters (water content, organic and inorganic matter) were determined. The abundance of phototrophic microorganisms was assessed on microscope slides which contained 1 mg of biofilm that was dehydrated for a short period of time and homogenized prior to slide preparation, and then rehydratated. The biomass of recorded cyanobacterial and algal taxa was calculated by applying geometric approximations and standard mathematical formulas. In Degurić Cave, at the sampling site where the highest biomass was documented, the higher diversity, water content and chlorophyll values were also recorded, while in Vernjikica Cave a high content of organic matter was documented. According to the multivariate analyses performed, the biomass of simple trichal Cyanobacteria, Bacillariophyta, and Xanthophyta was positively correlated with the content of organic matter in biofilm and light intensity, while coccoid and heterocytous Cyanobacteria and Chlorophyta showed a positive correlation with water content in the biofilm, relative humidity and distance from the entrance. The total biomass was positively correlated with the chlorophyll content, organic matter and light intensity, and negatively with the distance from the entrance