Convolution Properties of Certain Classes of Analytic Functions Defined by Jackson q-Derivative

In this paper, the Jackson q-derivative is used to investigate two classes of analytic functions in the open unit disc. The coefficient conditions and inclusion properties of the functions in these classes are established by convolution methods.

On the Notion of Reproducibility and Its Full Implementation to Natural Exponential Families

Let F=Fθ:θ∈Θ⊂R be a family of probability distributions indexed by a parameter θ and let X1,⋯,Xn be i.i.d. r.v.’s with L(X1)=Fθ∈F. Then, F is said to be reproducible if for all θ∈Θ and n∈N, there exists a sequence (αn)n≥1 and a mapping gn:Θ→Θ,θ⟼gn(θ) such that L(αn∑i=1nXi)=Fgn(θ)∈F. In this paper, we prove that a natural exponential family F is reproducible iff it possesses a variance function which is a power function of its mean. Such a result generalizes that of Bar-Lev and Enis (1986, The Annals of Statistics) who proved a similar but partial statement under the assumption that F is steep as and under rather restricted constraints on the forms of αn and gn(θ). We show that such restrictions are not required. In addition, we examine various aspects of reproducibility, both theoretically and practically, and discuss the relationship between reproducibility, convolution and infinite divisibility. We suggest new avenues for characterizing other classes of families of distributions with respect to their reproducibility and convolution properties .

Certain subclass of p-valent functions associated with Bessel functions

We introduce the subclass Sjp (?; c, k; ?), of p-valent functions associated with Bessel functions. Such results as inclusion relationships, convolution properties for this class are proved, coefficient estimates and certain integral preserving properties are also established with this class.

Conformable differential operators for meromorphically multivalent functions

We define a conformable diff-integral operator for a class of meromorphically multivalent functions. We show that this conformable operator adheres to the semigroup property. We then use the subordination properties to prove inclusion conditions, sufficienrt inclusion conditions and convolution properties for this class of conformable operators.

Shale Digital Core Image Generation Based on Generative Adversarial Networks

Stochastic reconstruction of digital core images is a vital part of digital core physics analysis, aiming to generate representative microstructure samples for sampling and uncertainty quantification analysis. This paper proposes a novel reconstruction method of the digital core of shale based on generative adversarial networks (GANs) with powerful capabilities of the generation of samples. GANs are a series of unsupervised generative artificial intelligence models that take the noise vector as an input. In this paper, the GANs with a generative and a discriminative network are created respectively, and the shale image with 45 nm/pixel preprocessed by the three-value-segmentation method is used as training samples. The generative network is used to learn the distribution of real training samples, and the discriminative network is used to distinguish real samples from synthetic ones. Finally, realistic digital core samples of shale are successfully reconstructed through the adversarial training process. We used the Fréchet inception distance (FID) and Kernel inception distance (KID) to evaluate the ability of GANs to generate real digital core samples of shale. The comparison of the morphological characteristics between them, such as the ratio of organic matter and specific surface area of organic matter, indicates that real and reconstructed samples are highly close. The results show that deep convolutional generative adversarial networks with full convolution properties can reconstruct digital core samples of shale effectively. Therefore, compared with the classical methods of reconstruction, the new reconstruction method is more promising.

Janowski Type q-Convex and q-Close-to-Convex Functions Associated with q-Conic Domain

Certain new classes of q-convex and q-close to convex functions that involve the q-Janowski type functions have been defined by using the concepts of quantum (or q-) calculus as well as q-conic domain Ω k , q [ λ , α ] . This study explores some important geometric properties such as coefficient estimates, sufficiency criteria and convolution properties of these classes. A distinction of new findings with those obtained in earlier investigations is also provided, where appropriate.