Search Results (5)

The research on coefficient inequalities in various classes of univalent holomorphic functions focuses on interpreting their coefficients through the coefficients associated with Carathéodory functions. Therefore, researchers can investigate the behavior of coefficient functionals by applying the known inequalities for Carathéodory functions. This study will explore various coefficient inequalities employing the techniques developed for the previously discussed family of functions. These coefficient inequalities include the Krushkal, Zalcman, and Fekete-Szegö inequalities, along with the second and third Hankel determinants. The class of symmetric starlike functions linked with a petal-shaped domain is the primary focus of our study. Full article

In the past, many conventional algorithms, such as self-adaptive dynamic differential evolution and asynchronous particle swarm optimization, were used to reconstruct buried objects in the frequency domain; these were unfortunately time-consuming during the iterative, repeated computing process of the scattered field. Consequently, we propose an innovative deep convolutional neural network approach to solve the electromagnetic inverse scattering problem for buried conductors in this paper. Different shapes of conductors are buried in one half-space and the electromagnetic wave from the other half-space is incident. The shape of the conductor can be reconstructed promptly by inputting the received scattered fields measured from the upper half-space into the deep convolutional neural network module, which avoids the computational complexity of Green’s function for training. Numerical results show that the root mean square error for differently shaped—circular, elliptical, arrow, peanut, four-petal, and three-petal—reconstructed images are, respectively, 2.95%, 3.11%, 17.81%, 15.10%, 14.14%, and 15.24%. Briefly speaking, not only can circular and elliptical buried conductors be reconstructed; some irregular shapes can be reconstructed well. On the contrary, the reconstruction result by U-Net for buried objects is worse since it is not able to obtain a good preliminary image by processing only the upper scattered field—that is, rather than the full space. In other words, our proposed deep convolutional neural network can efficiently solve the electromagnetic inverse scattering problem of buried conductors and provide a novel method for the microwave imaging of the buried conductors. This is the first successful attempt at using deep convolutional neural networks for buried conductors in the frequency domain, which may be useful for practical applications in various fields such as the medical, military, or industrial fields, including magnetic resonance imaging, mine detection and clearance, non-destructive testing, gas or wire pipeline detection, etc. Full article

The purpose of this article is to obtain the sharp estimates of the first four initial logarithmic coefficients for the class ${\mathcal{BT}}_s$ of bounded turning functions associated with a petal-shaped domain. Further, we investigate the sharp estimate of Fekete-Szegö inequality, Zalcman inequality on the logarithmic coefficients and the Hankel determinant $H_{2,1}\left(F_f/2\right)$ and $H_{2,2}\left(F_f/2\right)$ for the class ${\mathcal{BT}}_s$ with the determinant entry of logarithmic coefficients. Full article

In this article, we introduce a new class of multivalent analytic functions associated with petal-shape region. Furthermore, some useful properties, such as the Fekete–Szegö inequality, and their consequences for some special cases are discussed. For some specific value of function f, we obtain sufficient conditions for multivalent starlike functions connected with petal-shape domain. Finally, in the concluding section, we draw the attention of the interested readers toward the prospect of studying the basic or quantum (or q-) generalizations of the results, which are presented in this paper. However, the $(\mathfrak{p},q)$-variations of the suggested q-results will provide a relatively minor and inconsequential development because the additional (rather forced-in) parameter $\mathfrak{p}$ is obviously redundant. Full article

Flower colour and colour patterns are crucial traits for ornamental species; thus, a comprehensive understanding of their genetic basis is extremely significant for plant breeders. The tulip tree (Liriodendron tulipifera Linn.) is well known for its flowers, odd leave shape and tree form. However, the genetic basis of its colour inheritance remains unknown. In this study, a putative plastid terminal oxidase gene (LtuPTOX) was identified from L. tulipifera based on multiple databases of differentially expressed genes at various developmental stages. Then, the full-length cDNA of LtuPTOX was derived from tepals and leaves using RACE (rapid amplification of cDNA ends) approaches. Furthermore, gene structure and phylogenetic analyses of PTOX as well as AOXs (alternative oxidases), another highly similar homologue in the AOX family, were used to distinguish between the two subfamilies of genes. In addition, transient transformation and qPCR methods were used to determine the subcellular localization and tissue expression pattern of the LtuPTOX gene. Moreover, the expression of LtuPTOX as well as pigment contents was investigated to illustrate the function of this gene during the formation of orange bands on petals. The results showed that the LtuPTOX gene encodes a 358-aa protein that contains a complete AOX domain (PF01786). Accordingly, the Liriodendron PTOX and AOX genes were identified as only paralogs since they were rather similar in sequence. LtuPTOX showed chloroplast localization and was expressed in coloured organs such as petals and leaves. Additionally, an increasing pattern of LtuPTOX transcripts leads to carotenoid accumulation on the orange-band during flower bud development. Taken together, our results suggest that LtuPTOX is involved in petal carotenoid metabolism and orange band formation in L. tulipifera. The identification of this potentially involved gene will lay a foundation for further uncovering the genetic basis of flower colour in L. tulipifera. Full article