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The pioneering work in finance by Black, Scholes and Merton during the 1970s led to the emergence of the Black-Scholes (B-S) equation, which offers a concise and transparent formula for determining the theoretical price of an option. The establishment of the B-S equation, however, relies on a set of rigorous assumptions that give rise to several limitations. The non-local property of the fractional derivative (FD) and the identification of fractal characteristics in financial markets have paved the way for the introduction and rapid development of fractional calculus in finance. In comparison to the classical B-S equation, the fractional B-S equations (FBSEs) offer a more flexible representation of market behavior by incorporating long-range dependence, heavy-tailed and leptokurtic distributions, as well as multifractality. This enables better modeling of extreme events and complex market phenomena, The fractional B-S equations can more accurately depict the price fluctuations in actual financial markets, thereby providing a more reliable basis for derivative pricing and risk management. This paper aims to offer a comprehensive review of various FBSEs for pricing European options, including associated solution techniques. It contributes to a deeper understanding of financial model development and its practical implications, thereby assisting researchers in making informed decisions about the most suitable approach for their needs.

A comb structure consists of a one-dimensional backbone with lateral branches. These structures have widespread application in medicine and biology. Such a structure promotes an anomalous diffusion process along the backbone (x-direction), along with classical diffusion along the branches (y-direction). In this work, we propose a distributed-order time- and space-fractional diffusion-wave equation to model a comb structure in the more general setting. The distributed-order time- and space-fractional diffusion-wave equation is firstly formulated to study the anomalous diffusion in the comb model subject to an irregular convex domain with the motivation that the time-fractional derivative considers the memory characteristic and the space one with the variable diffusion coefficient possesses the nonlocal characteristic. The finite element method is applied to obtain the numerical solution. The stability and convergence of the numerical discretization scheme are derived and analyzed. Two numerical examples of relevance to the comb model are given to verify the correctness of the numerical method. Moreover, the influence of the involved parameters on the three-dimensional and axial projection drawing particle distribution subject to an elliptical domain are analyzed, and the physical meanings are interpreted in detail.

Asarum sieboldii Miq. is a leading economic crop and a traditional medicinal herb in China. Leaf-blade and petiole are the only aerial tissues of A. sieboldii during the vegetative growth, playing a vital role in the accumulation and transportation of biomass energy. They also act as critical indicators of drought in agricultural management, especially for crops having underground stems. During drought, variations in the morphology and gene expression of the leaves and petioles are used to control agricultural irrigation and production. Besides, such stress can also alter the differential gene expression in these tissues. However, little is known about the drought-tolerant character of the aerial parts of A. sieboldii. In this study, we examined the physiological, biochemical and transcriptomic responses to the drought stress in the leaf blades and petioles of A. sieboldii. The molecular mechanism, involving in drought stress response, was elucidated by constructing the cDNA libraries and performing transcriptomic sequencing. Under drought stress, a total of 2912 and 2887 unigenes were differentially expressed in the leaf blade and petiole, respectively. The detection of many transcription factors and functional genes demonstrated that multiple regulatory pathways were involved in drought tolerance. In response to drought, the leaf blade and petiole displayed a general physiological character, a higher SOD and POD activity, a higher MDA content and lower chlorophyll content. Three unigenes encoding POD were up-regulated, which can improve POD activity. Essential oil in petiole was extracted. The relative contents of methyleugenol and safrole in essential oil were increased from 0.01% to 0.05%, and 3.89% to 16.97%, respectively, while myristicin slightly reduced from 24.87% to 21.52%. Additionally, an IGS unigene, involved in eugenol biobiosynthesis, was found up-regulated under drought stress, which was predicated to be responsible for the accumulation of methyleugenol and safrole. Simple sequence repeats (SSRs) were characterized in of A. sieboldii, and a total of 5466 SSRs were identified. Among them, mono-nucleotides were the most abundant repeat units, accounting for 44.09% followed by tri-, tetra-, penta and hexa-nucleotide repeats. Overall, the present work provides a valuable resource for the population genetics studies of A. sieboldii. Besides, it provides much genomic information for the functional dissection of the drought-resistance in A. sieboldii, which will be useful to understand the bio-regulatory mechanisms linked with drought-tolerance to enhance its yield.

Asarum sieboldii Miq., a perennial herb in the family Aristolochiaceae, is widely used to treat colds, fever, headache and toothache in China. However, little is known about the drought-tolerance characteristics of A. sieboldii. In this study, to elucidate the molecular–genetic mechanisms of drought-stress tolerance of A. sieboldii, RNA-seq was conducted. In total, 53,344 unigenes were assembled, and 28,715 unigenes were annotated. A total of 6444 differential-expression unigenes (DEGs) were found, which were mainly enriched in phenylpropanoid, starch and sucrose metabolic pathways. Drought stress revealed significant up-regulation of the unigenes encoding PAL, C4H, HCT, C3H, CCR and IGS in the methyleugenol-biosynthesis pathway. Under the condition of maintaining drought for 15 days and 30 days, drought stress reduced the biosynthesis of volatile oil by 24% and 38%, respectively, while the production of key medicinal ingredients (such as methyl eugenol) was increased. These results provide valuable information about the diverse mechanisms of drought resistance in the A. sieboldii, and the changes in the expression of the genes involved in methyleugenol biosynthesis in response to drought stress.

This paper investigates the incompressible fractional MHD Maxwell fluid due to a power function accelerating plate with the first order slip, and the numerical analysis on the flow and heat transfer of fractional Maxwell fluid has been done. Moreover the deformation motion of fluid micelle is simply analyzed. Nonlinear velocity equation are formulated with multi-term time fractional derivatives in the boundary layer governing equations, and convective heat transfer boundary condition and viscous dissipation are both taken into consideration. A newly finite difference scheme with L1-algorithm of governing equations are constructed, whose convergence is confirmed by the comparison with analytical solution. Numerical solutions for velocity and temperature show the effects of pertinent parameters on flow and heat transfer of fractional Maxwell fluid. It reveals that the fractional derivative weakens the effects of motion and heat conduction. The larger the Nusselt number is, the greater the heat transfer capacity of fluid becomes, and the temperature gradient at the wall becomes more significantly. The lower Reynolds number enhances the viscosity of the fluid because it is the ratio of the viscous force and the inertia force, which resists the flow and heat transfer.

Asarum sieboldii Miq., one of the three original plants of TCM ASARI RADIX ET RHIZOMA, is a perennial herb distributed in central and eastern China, the Korean Peninsula, and Japan. Methyleugenol has been considered as the most important constituent of Asarum volatile oil, meanwhile asarinin is also employed as the quality control standard of ASARI RADIX ET RHIZOMA in Chinese Pharmacopeia. They both have shown wide range of biological activities. However, little was known about genes involved in biosynthesis pathways of either methyleugenol or asarinin in Asarum plants. In the present study, we performed de novo transcriptome analysis of plant tissues (e.g., roots, rhizomes, and leaves) at different developmental stages. The sequence assembly resulted in 311,597 transcripts from these plant materials, among which 925 transcripts participated in ‘secondary metabolism’ with particularly up to 20.22% of them falling into phenylpropanoid biosynthesis pathway. The corresponding enzymes belong to seven families potentially encoding phenylalanine ammonia-lyase (PAL), trans-cinnamate 4-monooxygenase (C4H), p-coumarate 3-hydroxylase (C3H), caffeoyl-CoA O-methyltransferase (CCoAOMT), cinnamoyl-CoA reductase (CCR), cinnamyl alcohol dehydrogenase (CAD), and eugenol synthase (EGS). Moreover, 5 unigenes of DIR (dirigent protein) and 11 unigenes of CYP719A (719A subfamily of cytochrome P450 oxygenases) were speculated to be involved in asarinin pathway. Of the 15 candidate CADs, four unigenes that possessed high FPKM (fragments per transcript kilobase per million fragments mapped) value in roots were cloned and characterized. Only the recombinant AsCAD5 protein efficiently converted p-coumaryl, coniferyl, and sinapyl aldehydes to their corresponding alcohols, which are key intermediates employed not only in biosynthesis of lignin but also in that of methyleugenol and asarinin. qRT-PCR revealed that AsCAD5 had a high expression level in roots at three developmental stages. Our study will provide insight into the potential application of molecular breeding and metabolic engineering for improving the quality of TCM ASARI RADIX ET RHIZOMA.

Aromatic polyketides from marine actinomycetes have received increasing attention due to their unusual structures and potent bioactivities. Compared to their terrestrial counterparts, marine aromatic polyketides have been less discovered and their structural and biological diversities are far from being fully investigated. In this study, we employed a PCR-based genome mining method to discover aromatic polyketides in our marine bacteria collection. Five new atypical angucyclinones, fluostatins M–Q (1–5) featuring a unique 6-5-6-6 ring skeleton, were discovered from one “positive” Streptomyces sp. PKU-MA00045. The structures of fluostatins M–Q (1–5) were elucidated based on comprehensive spectroscopic analyses and the crystallographic structure of fluostatin P (4), which contains the most oxidized A-ring, was solved by X-ray diffraction analysis with Cu Kα radiation. Compared to the published 16 fluostatin analogues, fluostatins M–Q (1–5) contained a different methoxy group attached at C-7 and hydroxy group attached at C-4, enriching the structural diversity of aromatic polyketides from marine actinomycetes. Genome sequencing of Streptomyces sp. PKU-MA00045 revealed the biosynthetic gene cluster of fluostatins M–Q (1–5), which contained different genes and gene organizations compared to known fluostatin gene clusters, facilitating the investigation of the biosynthesis of the unique 6-5-6-6 ring skeleton in all fluostatins.

Nonribosomal peptides from marine Bacillus strains have received considerable attention for their complex structures and potent bioactivities. In this study, we carried out PCR-based genome mining for potential nonribosomal peptides producers from our marine bacterial library. Twenty-one “positive” strains were screened out from 180 marine bacterial strains, and subsequent small-scale fermentation, HPLC and phylogenetic analysis afforded Bacillus sp. PKU-MA00092 and PKU-MA00093 as two candidates for large-scale fermentation and isolation. Ten nonribosomal peptides, including four bacillibactin analogues (1–4) and six bacillomycin D analogues (5–10) were discovered from Bacillus sp. PKU-MA00093 and PKU-MA00092, respectively. Compounds 1 and 2 are two new compounds and the 1H NMR and 13C NMR data of compounds 7 and 9 is first provided. All compounds 1–10 were assayed for their cytotoxicities against human cancer cell lines HepG2 and MCF7, and the bacillomycin D analogues 7–10 showed moderate cytotoxicities with IC50 values from 2.9 ± 0.1 to 8.2 ± 0.2 µM. The discovery of 5–10 with different fatty acid moieties gave us the opportunity to reveal the structure-activity relationships of bacillomycin analogues against these human cancer cell lines. These results enrich the structural diversity and bioactivity properties of nonribosomal peptides from marine Bacillus strains.

Texture enhancement is one of the most important techniques in digital image processing and plays an essential role in medical imaging since textures discriminate information. Most image texture enhancement techniques use classical integral order differential mask operators or fractional differential mask operators using fixed fractional order. These masks can produce excessive enhancement of low spatial frequency content, insufficient enhancement of large spatial frequency content, and retention of high spatial frequency noise. To improve upon existing approaches of texture enhancement, we derive an improved Variable Order Fractional Centered Difference (VOFCD) scheme which dynamically adjusts the fractional differential order instead of fixing it. The new VOFCD technique is based on the second order Riesz fractional differential operator using a Lagrange 3-point interpolation formula, for both grey scale and colour image enhancement. We then use this method to enhance photographs and a set of medical images related to patients with stroke and Parkinson's disease. The experiments show that our improved fractional differential mask has a higher signal to noise ratio value than the other fractional differential mask operators. Based on the corresponding quantitative analysis we conclude that the new method offers a superior texture enhancement over existing methods.

Fractional Fokker-Planck equations (FFPEs) have gained much interest recently for describing transport dynamics in complex systems that are governed by anomalous diffusion and nonexponential relaxation patterns. However, effective numerical methods and analytic techniques for the FFPE are still in their embryonic state. In this paper, we consider a class of time-space fractional Fokker-Planck equations with a nonlinear source term (TSFFPENST), which involve the Caputo time fractional derivative (CTFD) of order 𝛼∈ (0, 1) and the symmetric Riesz space fractional derivative (RSFD) of order 𝜇∈ (1, 2]. Approximating the CTFD and RSFD using the L1-algorithm and shifted Grünwald method, respectively, a computationally effective numerical method is presented to solve the TSFFPE-NST. The stability and convergence of the proposed numerical method are investigated. Finally, numerical experiments are carried out to support the theoretical claims.