Let E be a cookie-cutter set with dimH E =s. It is known that the Hausdorff s-measure and the packing s-measure of the set E are positive and finite. In this paper, we prove that for a gauge function g the set E has p...Let E be a cookie-cutter set with dimH E =s. It is known that the Hausdorff s-measure and the packing s-measure of the set E are positive and finite. In this paper, we prove that for a gauge function g the set E has positive and finite Hausdorff g-measure if and only if 0 〈 liminft→0 g（t）/ts 〈 ∞. Also, we prove that for a doubling gauge function g the set E has positive and finite packing g-measure if and only if 0 〈 lim supt→0 g（t）/ts 〈 ∞.展开更多
The characteristics of cookie-cutter sets in R^d are investigated. A Bowen's formula for the Hausdorff dimension of a cookie-cutter set in terms of the pressure function is derived. The existence of self-similar m...The characteristics of cookie-cutter sets in R^d are investigated. A Bowen's formula for the Hausdorff dimension of a cookie-cutter set in terms of the pressure function is derived. The existence of self-similar measures, conformal measures and Gibbs measures on cookie-cutter sets is proved. The dimension spectrum of each of these measures is analyzed. In addition, the locally uniformly a-dimensional condition and the fractal Plancherel Theorem for these measures are shown. Finally, the existence of order-two density for the Hausdorff measure of a cookie-cutter set is proved.展开更多
According to the bounded variation theory and the bounded distortion property of cookie-cutter-like （CCL） sets, the approximate self-similarity of cookie-cutter-like sets satisfying certain conditions is studied. Ba...According to the bounded variation theory and the bounded distortion property of cookie-cutter-like （CCL） sets, the approximate self-similarity of cookie-cutter-like sets satisfying certain conditions is studied. Based on the mean value theorem, it is proved that a class of special cookie-cutter-like sets is approximately self-similar. The results obtained in this paper extend the corresponding results that have already existed. K展开更多
Due to the numerous variables to take into account as well as the inherent ambiguity and uncertainty,evaluating educational institutions can be difficult.The concept of a possibility Pythagorean fuzzy hypersoft set(pP...Due to the numerous variables to take into account as well as the inherent ambiguity and uncertainty,evaluating educational institutions can be difficult.The concept of a possibility Pythagorean fuzzy hypersoft set(pPyFHSS)is more flexible in this regard than other theoretical fuzzy set-like models,even though some attempts have been made in the literature to address such uncertainties.This study investigates the elementary notions of pPyFHSS including its set-theoretic operations union,intersection,complement,OR-and AND-operations.Some results related to these operations are also modified for pPyFHSS.Additionally,the similarity measures between pPyFHSSs are formulated with the assistance of numerical examples and results.Lastly,an intelligent decision-assisted mechanism is developed with the proposal of a robust algorithm based on similarity measures for solving multi-attribute decision-making(MADM)problems.A case study that helps the decision-makers assess the best educational institution is discussed to validate the suggested system.The algorithmic results are compared with the most pertinent model to evaluate the adaptability of pPyFHSS,as it generalizes the classical possibility fuzzy set-like theoretical models.Similarly,while considering significant evaluating factors,the flexibility of pPyFHSS is observed through structural comparison.展开更多
This study identified castor oil and phosphate ester as effective retarders through setting time,tensile,and flexural tests,and determined their optimal dosages.The mechanism by which phosphate ester affects the setti...This study identified castor oil and phosphate ester as effective retarders through setting time,tensile,and flexural tests,and determined their optimal dosages.The mechanism by which phosphate ester affects the setting time of polyurethane was further investigated using molecular dynamics simulations.Fourier transform infrared spectroscopy was also employed to systematically study the physical and chemical interactions between phosphate esters and polyurethane materials.The results demonstrate that a 1%concentration of phosphate ester provides the most effective retarding effect with minimal impact on the strength of polyurethane.When phosphate ester is added to the B component of the two-component polyurethane system,its interaction energy with component A decreases,as do the diffusion coefficient and aggregation degree of component B on the surface of component A.This reduction in interaction slows the setting time.Additionally,the addition of phosphate ester to polyurethane leads to the disappearance or weakening of functional groups,indicating competitive interactions within the phosphate ester components that inhibit the reaction rate.展开更多
基金Supported by NSFC (Grant Nos.10571063,10771164)HuBei JiaoYuTing (Grant No.D20061001)
文摘Let E be a cookie-cutter set with dimH E =s. It is known that the Hausdorff s-measure and the packing s-measure of the set E are positive and finite. In this paper, we prove that for a gauge function g the set E has positive and finite Hausdorff g-measure if and only if 0 〈 liminft→0 g（t）/ts 〈 ∞. Also, we prove that for a doubling gauge function g the set E has positive and finite packing g-measure if and only if 0 〈 lim supt→0 g（t）/ts 〈 ∞.
基金supported by the National Natural Science Foundation of China.
文摘The characteristics of cookie-cutter sets in R^d are investigated. A Bowen's formula for the Hausdorff dimension of a cookie-cutter set in terms of the pressure function is derived. The existence of self-similar measures, conformal measures and Gibbs measures on cookie-cutter sets is proved. The dimension spectrum of each of these measures is analyzed. In addition, the locally uniformly a-dimensional condition and the fractal Plancherel Theorem for these measures are shown. Finally, the existence of order-two density for the Hausdorff measure of a cookie-cutter set is proved.
基金Supported by the National Natural Science Foundation of China (50872100)
文摘According to the bounded variation theory and the bounded distortion property of cookie-cutter-like （CCL） sets, the approximate self-similarity of cookie-cutter-like sets satisfying certain conditions is studied. Based on the mean value theorem, it is proved that a class of special cookie-cutter-like sets is approximately self-similar. The results obtained in this paper extend the corresponding results that have already existed. K
基金supported by the Deanship of Graduate Studies and Scientific Research at Qassim University(QU-APC-2024-9/1).
文摘Due to the numerous variables to take into account as well as the inherent ambiguity and uncertainty,evaluating educational institutions can be difficult.The concept of a possibility Pythagorean fuzzy hypersoft set(pPyFHSS)is more flexible in this regard than other theoretical fuzzy set-like models,even though some attempts have been made in the literature to address such uncertainties.This study investigates the elementary notions of pPyFHSS including its set-theoretic operations union,intersection,complement,OR-and AND-operations.Some results related to these operations are also modified for pPyFHSS.Additionally,the similarity measures between pPyFHSSs are formulated with the assistance of numerical examples and results.Lastly,an intelligent decision-assisted mechanism is developed with the proposal of a robust algorithm based on similarity measures for solving multi-attribute decision-making(MADM)problems.A case study that helps the decision-makers assess the best educational institution is discussed to validate the suggested system.The algorithmic results are compared with the most pertinent model to evaluate the adaptability of pPyFHSS,as it generalizes the classical possibility fuzzy set-like theoretical models.Similarly,while considering significant evaluating factors,the flexibility of pPyFHSS is observed through structural comparison.
基金Funded by the National Natural Science Foundation of China(No.52370128)the Fundamental Research Funds for the Central Universities(No.2572022AW54)。
文摘This study identified castor oil and phosphate ester as effective retarders through setting time,tensile,and flexural tests,and determined their optimal dosages.The mechanism by which phosphate ester affects the setting time of polyurethane was further investigated using molecular dynamics simulations.Fourier transform infrared spectroscopy was also employed to systematically study the physical and chemical interactions between phosphate esters and polyurethane materials.The results demonstrate that a 1%concentration of phosphate ester provides the most effective retarding effect with minimal impact on the strength of polyurethane.When phosphate ester is added to the B component of the two-component polyurethane system,its interaction energy with component A decreases,as do the diffusion coefficient and aggregation degree of component B on the surface of component A.This reduction in interaction slows the setting time.Additionally,the addition of phosphate ester to polyurethane leads to the disappearance or weakening of functional groups,indicating competitive interactions within the phosphate ester components that inhibit the reaction rate.
