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The fuzzy stochastic differential equations with Markovian switching are considered. First, under the $ \rm Carath\acute {e}odory $ Carath e ´ odory assumptions, the existence and uniqueness theorem for the aforementioned equations is given by means of stopping time techniques, discretization method, and the Gronwall-Bellman inequality. Then the boundedness of the second moment of the solutions for such equations is established. Furthermore, by using the averaging method and stochastic analysis, as well as the Bihari's inequality, this paper mainly demonstrates the asymptotic relationship between the solutions of the original equations and the corresponding averaged equations in terms of mean square and probability. Finally, the theoretical result of the averaging method is illustrated by an example. [ABSTRACT FROM AUTHOR]

This study investigates the initial value problem of high-order variable-order φ -Hilfer fractional implicit integro-differential equations. Due to the lack of the semigroup property in variable-order fractional integrals, solving these equations presents significant challenges. We introduce a novel approach that approximates variable-order fractional derivatives using a piecewise constant approximation method. This method facilitates an equivalent integral representation of the equations and establishes the Ulam stability criterion. In addition, we explore higher-order forms of fractional-order equations, thereby enriching the qualitative and stability results of their solutions. [ABSTRACT FROM AUTHOR]

In the present article, the matrix projective synchronization (MPS) and the inverse matrix projective synchronization (IMPS) have been analyzed with fractional‐order chaotic systems with uncertain terms. First, we theoretically discussed both types of synchronizations. To achieve both types of synchronization, several required conditions are derived. We also drew portraits of Lyapunov's exponents and bifurcation for more analysis. Furthermore, time‐delay terms are also considered, giving an elegant application in real‐world problems. Finally, numerical results agreed with the theoretical hypothesis. The numerical outcomes are shown through the graphical presentation. [ABSTRACT FROM AUTHOR]

Ichthyosis follicularis with atrichia and photophobia (IFAP) syndrome is a rare genodermatosis characterized by a classic triad of follicular ichthyosis, alopecia, and photophobia. We report a Chinese patient displaying features of IFAP triad along with painful palmoplantar keratoderma, recurrent infections, periorificial keratotic plaques, nail dystrophy, and pachyonychia. Whole‐exome sequencing revealed an intronic variant (NM_015884.3: exon7:c.970+5G>A) in the gene MBTPS2. Sanger sequencing confirmed that the variant segerated with phenotype in the family. Sequencing of cDNAs derived from the patient indicated the variant introduced a new splice donor site, leading to partial skipping of exon 7 (r.951_970del). An in vitro mini‐gene assay also revealed abnormal splicing of exon 7. This study presents a case complicated with X‐linked IFAP syndrome and Olmsted syndrome, and highlights the significance of using validation assays to identify the pathogenicity of intronic variants in MBTPS2. [ABSTRACT FROM AUTHOR]

The coordinated use of electricity and a heat energy system can effectively improve the energy structure during winter heating in the northern part of China and improve the environmental pollution problem. In this paper, an economic scheduling model of an electric–thermal integrated energy system, including a wind turbine, regenerative electric boiler, solar heat collection system, biomass boiler, ground source heat pump and battery is proposed, and a biomass boiler was selected as the auxiliary heat source of the solar heat collection system. A mixed integer linear programming model was established to take the operating cost of the whole system as the target. A day-ahead optimization scheduling strategy considering the demand side response and improving new energy consumption is proposed. In order to verify the influence of the coordinated utilization of the flexible load and energy storage equipment on the optimal scheduling in the model built, three scenarios were set up. Scenario 3 contains energy storage and a flexible load. Compared with scenario 1, the total cost of scenario 3 was reduced by 51.5%, and the abandonment cost of wind energy was reduced by 43.3%. The use of a flexible load and energy storage can effectively reduce the cost and improve new energy consumption. By increasing the capacity of the energy-storage device, the wind power is completely absorbed, but the operation and maintenance cost is increased, so the capacity of energy storage equipment is allocated reasonably according to the actual situation. [ABSTRACT FROM AUTHOR]