We investigate the existence of solution for a Fractional integro-differential inclusion via the Caputo-Hadamard Fractional derivation. We prove that Dimension of the set of solutions for the inclusion problem isinfinite Dimensional under some conditions.

Acne is a very common skin disease. Scars are seen in 95% of patients with acne. Although numerous treatments have been recommended to cure acne scars, researchers are still searching for a single modality to treat the complication due to its variety in shape and depth. In other words, a combination of available methods is required to reach satisfactory results. We compared the effects of Fractional carbon dioxide (CO2) laser alone and in combination with subcision.Methods: This clinical trial study was performed in Skin Diseases and Leishmaniasis Research Center (Isfahan, Iran) during 2011-12. Eligible patients with atrophic acne scars were treated with Fractional CO2 laser alone (five sessions with three-week interval) on right side of the face and Fractional CO2 laser plus subcision (one session using both followed by four sessions of Fractional CO2 laser, all with three-week intervals) on the left side. The subjects were visited one, two, and six months after the treatment. Patient satisfaction rate was analyzed by SPSS20.Findings: The average of recovery rate was 54.7% using the combination method and 43.0% using laser alone (P<0.001). The mean patient satisfaction rate according to visual analogue scale (VAS) score was significantly higher with the combination method than with laser alone (6.6 vs. 5.2; P<0.001). Bruising was only seen with the combination method and lasted for one week in 57.0% of patients and for two weeks in 43.0%. Erythema was a side effect of both methods. Post-inflammatory pigmentation and hyperpigmentation were only caused by the combination method. None of the patients had persistent side effects after six months.Conclusion: Using a combination of subcision and laser had suitable results regarding scar recovery and satisfaction rate. However, bruising, post-inflammatory pigmentation, and hyperpigmentation were the side effects of this method. In general, subcision plus laser can be beneficial in the treatment of atrophic acne scars.

There is no data on the effect of maximum and minimum premedication dose of atropine on work performance of heart and amount of Fractional Shortening (FS) in rabbit echocardiography, so this study was designed for evaluation of probability of the above effect.24 White New Zealand rabbits, clinically healthy with no consideration in sex and age, weighting 1.5±0.3 kg in four groups consisting six animals for access to the purpose of study were evaluated.Injection of atropine with maximum and minimum dosage for first and second groups and injection of distilled water for third and fourth groups were observed. Calculations of FS for left ventricles in statutory times were done by echocardiography with no knowledge about injection solution.Determinations for Mean, Standard Deviation, Minimum and Maximum amount for FS were 37.29, 3.355, 30 and 43 respectively. Determinations for Mean and Standard Deviation for FS in group one, before injection and in minutes 2, 5, 15 and 30 after injection were 36.33±3.011, 35.33±2.944, 36.17±4.119, 36.83±3.251 and 37.00±2.449 respectively. Determinations for Mean and Standard Deviation for FS in group two, before injection and in minutes 2, 5, 15 and 30 after injection were 39.17±2.994, 39.00±3.347, 36.50±3.017, 35.50±6.025 and 36.83±5.811 respectively. Determinations for Mean and Standard Deviation for FS in group three, before injection and in minutes 2, 5, 15 and 30 after injection were 36.17±2.483, 33.83±3.061, 34.33±2.805, 35.00±2.828 and 33.67±4.885 respectively. Determinations for Mean and Standard Deviation for FS in group four, before injection and in minutes 2, 5, 15 and 30 after injection were 37.50±4.550, 34.17±4.622, 37.17±3.125, 36.50±1.225 and 37.50±2.811 respectively.As a conclusion, Stability of heart performance was observed after using maximum dose of atropine.

Let G be a simple connected graph. In this paper, Szeged Dimension and PIv Dimension of graph G are introduced. It is proved that if G is a graph of Szeged Dimension 1 then line graph of G is 2-connected. Trees of Szeged Dimension 1 are characterized. The Szeged Dimension and PIv Dimension of five composite graphs: sum, corona, composition, disjunction and symmetric difference with strongly regular components are computed. Also explicit formulas of Szeged and PIv indices for these composite graphs are obtained.

Modeling the joint distribution of liquidity and return to determine the dependency structure of a 15-stock portfolio using intraday data in 2019 provides a suitable model for the commonalities. Based on the complexities of higher Dimensions multivariate modeling (combined distribution of liquidity and return), after univariate modeling of the stocks’,liquidity based on Autoregressive Conditional Poisson model (ACP) and the returns with the Generalized Autoregressive Conditional Heteroscedastic (GARCH), the marginal distributions were incorporated into a Copula Vine to model the dependency. The findings of this study, based on high-Dimensional high-frequency data, indicate that there is an extreme nonlinear correlation between the liquidity of stocks and also between liquidity and returns across the portfolio, which is necessary to take into account in risk assessments to prevent inaccurate assessment of risk indicators such as value at risk. In addition, the results have shown that modeling the joint distribution of liquidity and stock returns in high Dimensions relying on the Copula Vine model due to the flexibility performs well.