This paper describes the preparation and description of four new complexes [Bu₂Sn(AA)Cl] (AA = glycine, DL-valine and L-leucine) and [Bu₂SnPhen₂] (Phen = DL-phenylalanine) They were characterized by elemental analyses of carbon, hydrogen, nitrogen, chlorine and tin; and by infrared, ^119mSn-Mössbauer and ^119mSn-RMN spectroscopies, and by the applications of TG and DSC techniques in dynamic helium atmosphere. For the [Bu₂Sn(AA)Cl], trigonal bipyramidal tin species, AA were coordinated bidentatedly by the carboxylic oxygen and by the NH₂ group. The [Bu₂SnPhen₂)], with tin atoms in trans-octahedral sites, had the Phen-coordinated by both carboxylic oxygen atoms.

The microbial contamination in food packaging have been a major concern that paved the way for the search for natural based new anti-microbial agents, such as modified α-mangostin. In the present work, twelve synthetic analogs were obtained via semi-synthetic modification of α-mangostin by Ritter reaction, reduction by palladium-carbon (Pd-C), alkylation, and acetylation. The evaluation of the anti-microbial potential of the synthetic analogs showed higher therapeutic value than the parent molecule. The anti-microbial studies proved that I E showed higher antibacterial activity whereas I I showed most significant antifungal activity. Due to their microbial properties, modified α-mangostin can be utilized as active anti-microbial agents in food packaging.

Ferrocenium (Fc+) inherits a number of molecular/electronic properties from the neutral counterparts’ ferrocene (Fc) including the high symmetry. Both Fc+ and Fc prefer the eclipsed structure (D5h) over the staggered structure (D5d) by an energy of 0.36 kcal·mol-1. The present study using the recently developed excess orbital energy spectrum (EOES) shows that the open shell Fc+ cation exhibits similar conformer dependent configurational changes to the neutral Fc conformer pair. A further energy decomposition analysis (EDA) discloses that the reasons for the preferred structures are different between Fc+ and Fc. The dominant differentiating energy between the Fc+ conformers is the electrostatic energy (EEstat), whereas in neutral Fc, it is the quantum mechanical Pauli repulsive energy (EPauli). Within the D5h conformer of Fc+, the EOES reveals that the -electrons of Fc+ experience more substantial conformer dependent energy changes than the -electrons (assumed the hole is in a β orbital).

In this paper, we establish the generalized Qi-type inequality involving conformable fractional integrals. The results presented here would provide extensions of those given in earlier works.

In this paper, by means of the Banach fixed point theorem and the Krasnoselskii's fixed point theorem, we investigate the existence of solutions for some fractional neutral functional integro-differential equations involving infinite delay. This paper deals with the fractional equations in the sense of Caputo fractional derivative and in the Banach spaces. Our results generalize the previous works on this issue. Also, an analytical example is presented to illustrate our results.

Exact solutions to conformable time fractional (3+1)- dimensional equations are derived by using the modified form of the Kudryashov method. The compatible wave transformation reduces the equations to an ODE with integer orders. The predicted solution of the finite series of a rational exponential function is substituted into this ODE. The resultant polynomial equation is solved by using algebraic operations. The method works for the Jimbo-Miwa, the Zakharov-Kuznetsov and the modified Zakharov-Kuznetsov equations in conformable time fractional forms. All the solutions are expressed in explicit forms.

With the growing hope that disease-modifying treatments could target the molecular basis of neurodegenerative diseases even before the onset of symptoms, there is mounting pressure to define disease entities based on pathophysiology rather than on clinical syndromes. The Alzheimer’s disease research community has recently transitioned from diagnostic criteria based on an amnestic syndrome to a purely biomarker-based disease definition, relying on the demonstration of amyloid-beta pathology, tau pathology, and neurodegeneration. In contrast, current diagnostic criteria for Parkinson’s disease still rely on the presence of the well-described clinical syndrome of parkinsonism, with the addition of characteristic motor- and non-motor signs and symptoms. However, there is now unequivocal evidence that Parkinson’s disease starts years before the onset of parkinsonism. Furthermore, neuropathologically defined Lewy body disease is clinically heterogeneous, combining a range of motor, non-motor, dopaminergic and non-dopaminergic features. Finally, clinically defined Parkinson’s disease has diverse underlying etiologies most, but not all, associated with α-synuclein positive Lewy pathology. In light of recent scientific advances, we propose a biologically based definition for the diagnosis of Parkinson's disease, initially to be used for research purposes. The criteria use a three-component ‘G-S-N’ system. The first is documentation of defined gene variants (‘G’), which cause or strongly predispose to PD as the most upstream component. The second is α-synuclein pathology (‘S’), currently defined as pathological α-synuclein deposition in tissue or positive α-synuclein seeding assays. The third is evidence of underlying neurodegeneration (‘N’), currently defined by specific neuroimaging procedures. The associated clinical syndrome (‘C’) is defined by a single high-specificity feature or multiple lower-specificity features. Initiating this transition will enable the field to fuel both basic and clinical research and move closer to the precision medicine required to develop clinically meaningful disease-modifying therapies. We acknowledge current limitations, ethical implications, and the need for prospective validation of this approach.

In this paper, we have established Hermite-Hadamard inequalities for r-preinvex functions via fractional integrals.

The exact solutions of some conformable time fractional PDEs are presented explicitly. The modified Kudryashov method is applied to construct the solutions to the conformable time fractional Regularized Long Wave-Burgers (RLW-Burgers, potential Korteweg-de Vries (KdV) and clannish random walker's parabolic (CRWP) equations. Initially, the predicted solution in the finite series of a rational form of an exponential function is substituted to the ODE generated from the conformable time fractional PDE by using wave transformation. The coefficients used in the finite series are determined by solving the algebraic system derived from the coefficients of the powers of the predicted solution.

Local polynomial smoother (LPS) is a weighted local least-squares nonparametric method. It provides a local Taylor series fit of the data at any location and can be directly used in a differential equation to provide a numerical scheme. In this article, we introduce this new nonparametric idea based on local polynomial smoother, for acquiring the numerical solution of the Bagley-Torvik fractional-order differential equations. Furthermore, this paper will present a numerical comparison with some methods, such as legendre operational matrix and pseudo-spectral method. The efficiency and accuracy of the LPS method are demonstrated by two numerical examples.

The three dimensional conformable time fractional Kadomtsev-Petviashvili and the conformable time fractional modified Kawahara equations are solved by implementing the Kudryashov's procedure. The corresponding wave transformation reduces both equations to some ODEs. Balancing the nonlinear and the highest order derivative terms gives the structure of the solutions in the finite series form. The useful symbolic tools are used to solve the resultant algebraic systems. The solutions are expressed in explicit forms.

In this paper we use the conformable fractional derivative to discuss some fractional linear differential equations with constant coefficients. By applying some similar arguments to the theory of ordinary differential equations, we establish a sufficient condition to guarantee the reliability of solving constant coefficient fractional differential equations by the conformable Laplace transform method. Finally, we analyze the analytical solution for a class of fractional models associated with Logistic model, Von Foerster model and Bertalanffy model is presented graphically for various fractional orders and solution of corresponding classical model is recovered as a particular case.

The mechanistic pathway of TEMPO/I₂-mediated oxidative cyclization of N,N-diaryl amino alcohols 1 is investigated in this study. Based on our direct empirical experiments, three key intermediates (the aminium radical cation 3, the α-aminoalkyl radical 4, and the iminium 5), four kind reactive species (radical TEMPO, cationic TEMPO, TEMPO-I and iodo radical) and three kind pathways (1. SET/PCET mechanism, 2. HAT/1,6-H transfer mechanism, 3. Ionic mechanism) were assumed. Under the assumption, nine free energy diagrams are acquired through DFT calculation. From the comparison of the solution phase free energy, some possibility can be excluded and then the chosen plausible mechanisms are concretized with the more stable intermediate 7.

In this paper, we investigate an analytical solution of a class of nonlinear fractional integro-differential equation base on a generalized fractional power series expansion. The fractional derivatives are described in the conformable's type. The new approach is a modified form of the well-known Taylor series expansion. The illustrative examples are presented to demonstrate the accuracy and effectiveness of the proposed method

The exact solutions in the wave form are derived for the time fractional KdV and the time fractional Burgers' equations in conformable fractional derivative sense. The fractional variable change using the fundamental properties of the conformable derivative reduces both equations to some nonlinear ODEs. The predicted solution is assumed to be a finite series form of a function satisfying a particular first-order ODE whose solution contains an exponential function in the denominator. The solutions are represented in the explicit forms and illustrated by some choices of the parameters for various fractional orders of the equations.

This paper analyzes and compares models for predicting average magnet losses in interior permanent-magnet motors with fractional-slot concentrated windings due to harmonics in the armature reaction (assuming sinusoidal phase currents). Particularly, loss models adopting different formulations and solutions to the Helmholtz equation to solve for the eddy currents are compared to a simpler model relying on an assumed eddy-current distribution. Boundaries in terms of magnet dimensions and angular frequency are identified (numerically and using an identified approximate analytical expression) to aid the machine designer whether the more simple loss model is applicable or not. The assumption of a uniform flux-density variation (used in the loss models) is also investigated for the case of V-shaped and straight interior permanent magnets. Finally, predicted volumetric loss densities are exemplified for combinations of slot and pole numbers common in automotive applications.

Passiflora cincinnata Mast is a plant popularly used in traditional medicine in northeastern Brazil. The present study aimed to evaluate the anti-lipoperoxidative and anti-α-amylase properties, and cytotoxicity of an extract and fraction of passion fruit leaves (P. cincinnata Mast), as well as the antihyperglecemiant activity of the fraction rich in glycosylated flavonoids and showing low cytotoxicity in rats with a postprandial hyperglycaemia condition. The ethyl acetate fraction (F.ACT) of the P. cincinnata leaves presented the best anti-lipoperoxide properties with TBARs 81.49%, 95.48% and 75.62% lower than AAPH, FeSO₄ and H₂O₂ induced controls, respectively, at 200 μg.mL-1. In addition, it presented an anti-α-amylase potential, with a better ability to inhibit the α-amylase enzyme in comparison to the acarbose control (IC₅₀ 6.49 ± 0.11 and 12.01 ± 0.4 μg.mL-1, respectively). The hydroalcoholic extract of P. cincinnata (ExEt) presented high content of total tannins, flavonoids and flavonols. However, F.ACT had the highest concentration of flavonoids among the fractions studied. HPLC analysis of this fraction revealed the presence of the flavonoids isovitexin, orientin and isoorientin. F.ACT showed low to no cytotoxicity below 150 μg.mL-1. Regarding the post-prandial antihyperglycaemic activity of F.ACT, it was observed at 50 and 100 mg.kg-1.

The Sine-Gordon expansion method is implemented to construct exact solutions some conformable time fractional equations in Regularized Long Wave(RLW)-class. Compatible wave transform reduces the governing equation to classical ordinary differential equation. The homogeneous balance procedure gives the order of the predicted polynomial-type solution that is inspired from well-known Sine-Gordon equation. The substitution of this solution follows the previous step. Equating the coefficients of the powers of predicted solution leads a system of algebraic equations. The solution of resultant system for coefficients gives the necessary relations among the parameters and the coefficients to be able construct the solutions. Some solutions are simulated for some particular choices of parameters.

In this paper, we study the fractional pseudo-parabolic equations u_t + (−△)^s u + (−△)^s u_t = u log |u|. Firstly, we recall the relationship between the fractional Laplace operator (−△)^s and the fractional Sobolev space H^s and discuss the invariant sets and the vacuum isolating behavior of solutions with the help of a family of potential wells. Then, we derive a threshold result of existence of global weak solution: for the low initial energy J(u₀) < d, the solution is global in time with I(u₀) > 0 or ∥u₀∥X_0(Ω) = 0 and blows up +∞ with I(u₀) < 0; for the critical initial energy J(u₀) = d, the solution is global in time with I(u₀) ≥ 0 and blows up at +∞ with I(u₀) < 0. The decay estimate of the energy functional for the global solution is also given.

Polyunsaturated fatty acids (PUFAs) and antioxidants are important mediators in the central nervous system (CNS). Lipid derivatives may be used to generate endocannabinoids or prostanoids derived from arachidonic acid, which attenuates excitotoxicity in quadripartite synapses with a focus in astrocytes and microglia; on the other hand, antioxidants, such as glutathione (GSH) and ascorbate, have been shown to signal through transmitter receptors and protect against acute and chronic oxidative stress, modulating the activity of different signaling pathways. Several authors have investigated the role of these nutrients in young and senescent brain, as well as in degenerative conditions such as Alzheimer’s and Parkinson's diseases. Through literature review, we aimed to highlight recent data on the role of fatty acids, antioxidants and physical activity in physiology and in molecular mechanisms of brain senescence. Data indicate the complexity and essentiality of endogenous/dietary antioxidants for maintenance of the redox status and control of neuroglial signaling under stress. Recent studies also indicate that omega-3 and -6 fatty acids act in a competitive manner to generate mediators for energy metabolism, feeding behavior, plasticity and memory mechanisms throughout aging. Finding pharmacological or dietary resources that mitigate or prevent neurodegenerative affections continues to be a great challenge and require additional efforts from researchers, clinicians and nutritionists in the field.

The key objective of the present paper is to propose a numerical scheme based on the homotopy analysis transform technique to analyze the time-fractional nonlinear predator-prey population model. The population model is coupled fractional order nonlinear partial differential equations often employed to narrate the dynamics of biological systems in which two species interact, first is a predator and the second is a prey. The proposed scheme provides the series solution with great freedom and flexibility by choosing appropriate parameters. The convergence of results is free from small or large parameters. Three examples are discussed to demonstrate the correctness and efficiency of the used computational approach.

A metastable β-Ti alloy, Ti–10V–3Fe–3Al (wt.%), was subjected to thermos-mechanical processing including the compression test at 725°C, which is below the β transus temperature (780°C), and at strain rate of 10-3s-1. The presence of phases was determined using transmission electron microscopy and X-ray diffraction. Although the dynamic recovery took place together with slip, both deformation-induced α˝ martensite and ω were detected as other operating mechanisms for the first time in metastable-β Ti alloys deformed in α+β region. The volume fraction of stress-induced α˝ was higher than that of the same alloy deformed at room temperature due to higher strain applied. Stress-induced twinning was not operational, which could be related to the priority of slip mechanism at high temperature resulted from thermally-assisted nucleation and lateral migration of kink-pairs.

We have drawn a few interesting conclusions while studying reaction products of Ph₂Si(OH)₂ with Al(ⁱBu)₃ and tetraisobutylalumoxane. In the first place, this is the production (at Ph₂Si(OH)₂ and Al(ⁱBu)₃ equimolar ratio) of oligomer siloxyalumoxane structure with alternating four- and six-member rings, as well as isobutyl and phenyl groups migration between aluminum and silicon due to the formation of intramolecular four-member cyclic complex [Ph₂(OH)SiO]Al(ⁱBu)₂ → [(ⁱBu)Ph(OH)SiO]Al(ⁱBu)Ph. Ph₂Si(OH)₂ interaction with Al(ⁱBu)₃ not only starts from intramolecular complex production, but the chain is terminated for the same reason, which in the case of Ph₂Si(OH)₂ reaction with tetraisobutylalumoxane results in failure of obtaining high-polymer siloxyalumoxane compounds. When Al(ⁱBu)₃ interacts with α- and γ-diols, no oligomer compounds are produced. Al(ⁱBu)₃ reaction with α, γ-diols results in monomer compounds that are likely to have cyclic structure. Notably at Al(ⁱBu)₃ interaction with α-diol only double excess of Al(ⁱBu)₃ allows full replacement of hydrogen in α-diol hydroxyl groups by aluminum alkyl residue with 1,3-bis(diisobutylalumoxymethyl)-1,1,3,3-tetramethyldisiloxane production. At equimolar ratio of initial reagents the second isobutyl radical at Al does not interact with the second hydroxyl group of α-diol, apparently due to the steric hindrance and 1-(diisobutylalumoxymethyl)-3-(hydroxymethyl)-1,1,3,3-tetramethyl-disiloxane is produced. Al(ⁱBu)₃ reactions with γ-diol also result in monomer compounds but the presence of a chain consisting of three СН₂-groups between Si and hydroxyl group facilitates interaction between the second hydroxyl group of γ-diol and the second isobutyl radical Al(ⁱBu)₃. Tetraisobutylalumoxane reaction with α- and γ-diols results in oligomer compounds.

We report the results of our study of approved drugs as potential treatments for COVID 19, based on the application of various bioinformatics predictive methods. The drugs studied include chloroquine, ivermectin, remdesivir, sofosbuvir, boceprevir, and α-difluoromethylornithine (DMFO). Our results indicate that these small molecules selectively bind to stable, kinetically active residues and residues adjoining them on the surface of proteins and inside protein pockets, and that some prefer hydrophobic over other active sites. Our approach is not restricted to viruses and can facilitate rational drug design, as well as improve our understanding of molecular interactions, in general.

This paper first discusses a fractional-order Liu system of order as low as 2.7 and shows its chaotic characteristics by carrying out numerical simulations such as Lyapunov exponents, bifurcation diagrams and phase portraits. Then, by using the topological horseshoe theory and computer-assisted proof, the existence of chaos in the system is verified theoretically. Finally, the fractional-order system is implemented on a Field Programmable Gate Array (FPGA) and the results obtained show that the fractional-order Liu system is indeed chaotic.

In this work α-synuclein amyloid fibrils, formation of which is a biomarker of the Parkinson’s disease, were investigated with the use of fluorescent probe thioflavin T (ThT). Experimental conditions of the protein fibrillogenesis were chosen so that a sufficient number of continuous measurements can be performed to characterize and analyze all stages of this process. The reproducibility of fibrillogenesis and the structure of the obtained aggregates (that is a critical point for their further investigation) were proved using a wide range of physical-chemical methods. For determination of ThT—α-synuclein amyloid fibrils binding parameters sample and reference solutions were prepared with the use of equilibrium microdialysis. By absorption spectroscopy of these solutions ThT—fibrils binding mode with the binding constant about 10⁴ M⁻¹ and stoichiometry of ThT per protein molecule about 1:8 was observed. Fluorescence spectroscopy of the same solutions with the subsequent correction of the recorded fluorescence intensity on the primary inner filter effect allowed to determine another mode of ThT binding to fibrils with the binding constant about 10⁶ M⁻¹ and stoichiometry about 1:2500. Analysis of photophysical characteristics of the dye molecules bound to the sites of different binding modes allowed to assume the possible localization of these sites. Obtained differences in the ThT binding parameters to amyloid fibrils formed from α-synuclein and other amyloidogenic proteins, as well as in the photophysical characteristics of the bound dye, confirmed the hypothesis of amyloid fibrils polymorphism.

Permanent magnet synchronous motor (PMSM) AC servo system shows the characteristics of uncertainty, time-varying and non-linearity, which makes it difficult for traditional PID control to achieve ideal control effect. Fuzzy control has strong adaptability to the problems of parameter variation, non-linearity and model inaccuracy of the controlled object. Vector control strategy is used to study its control principle and the realization of SVPWM. Because the motor object has certain fractional order characteristics, the fuzzy parameter self-tuning PIλ control is chosen as the position regulator of servo motor. It combines the accuracy of fractional order PI control and the adaptability of fuzzy control. A simulation model of PMSM three-closed-loop system is built in MATLAB/Simulink environment. The results show that the control method is effective and can satisfy the trajectory tracking of servo control.

The main objective of this paper is to introduce k-fractional derivative operator by using the definition of k-beta function. We establish some results related to the newly defined fractional operator such as Mellin transform and relations to k-hypergeometric and k-Appell's functions. Also, we investigate the k-fractional derivative of k-Mittag-Leffler and Wright hypergeometric functions.

The space time fractional Korteweg-de Vries equation, modified Equal Width equation and Benney-Luke Equations are solved by using simple hyperbolic tangent Ansatz method. A simple compatible wave transformation in one dimension is employed to reduce the governing equations to integer ordered ODEs. Then, the ansatz approximation is used to derive exact solutions. Some illustrative examples are presented for some particular choices of parameters and derivative orders.

This paper discusses the concept of fractional derivative with complex order from the application point of view. It is shown that a fractional derivative is hermitian, if and only if the order is real. The hermitian part of complex order derivatives behave like allpass filters with almost logarithmic phase.

In this paper, some new properties of the fundamental solution to the multi-dimensional space- and time-fractional diffusion-wave equation are deduced. We start with the Mellin-Barnes representation of the fundamental solution that was derived in the previous publications of the author. The Mellin-Barnes integral is used to get two new representations of the fundamental solution in form of the Mellin convolution of the special functions of the Wright type. Moreover, some new closed form formulas for particular cases of the fundamental solution are derived. In particular, we solve an open problem of representation of the fundamental solution to the two-dimensional neutral-fractional diffusion-wave equation in terms of the known special functions.

Fractional differential mathematical model unfolding the dynamics of the COVID-19 pandemic in India is presented and explored in this paper. The purpose of this study is to estimate the future outbreak of disease and potential control strategies using mathematical models in India as a whole country as well as in some of the states of the country. This model is calibrated based on reported cases of infections over the month of April 2020 in India. We have used iterative fractional complex transform method to find approximate solutions of the model having modified Riemann Liouville fractional differential operator. We have also carried out a comparative analysis between actual and estimated cumulative cases graphically, moreover, most sensitive parameters for basic reproduction number$(R_0)$ are computed and their effect on transmission dynamics of COVID-19 pandemic is investigated in detail.

In this present paper, we deal with the generalized (k, s)-fractional integral and differential operators recently defined by Nisar et al. and obtain some generalized (k, s)-fractional integral and differential formulas involving the class of a function as its kernels. Also, we investigate a certain number of their consequences containing the said function in their kernels.

Recently, different extensions of the fractional derivative operator are found in many research papers. The main aim of this paper is to establish an extension of the extended Caputo fractional derivative operator. The extension of an extended fractional derivative of some elementary functions derives by considering an extension of beta function which includes the Mittag-Leffler function in the kernel. Further, an extended fractional derivative of some familiar special functions, the Mellin transforms of newly defined Caputo fractional derivative operator and the generating relations for extension of extended hypergeometric functions also presented in this study.

New exact solutions to conformable time fractional EW and modified EW equations are constructed by using Sine-Gordon expansion approach. The fractional traveling wave transform and homogeneous balance have significant roles in the solution procedure. The predicted solution is of the form of some finite series of multiplication of powers of cos and sin functions. The relation among trigonometric and hyperbolic functions in sense of Sine-Gordon expansion gives opportunity to construct the solutions in terms of hyperbolic functions.

In the present study, nonlinear vibration of a nanobeam resting on fractional order viscoelastic Winkler-Pasternak foundaion is studied using nonlocal elasticity theory. D'Alembert principle is used to derive the governing equation and the associated boundary conditions. The approximate analytical solution is obtained by applying the multiple scales method. Detailled parametric study is conducted, the effects of variation in different parameters belonging to the application problems on the system are calculated numerically and depicted. We remark that the order and the coefficient of the fractional derivative have significant effect on the natural frequency and the amplitude of vibrations

Based on the fractional order of nonlinear system for love model with a periodic function as an external force, analyzed the characteristics of the chaotic dynamic in this study. The relationship between the chaotic dynamic of the fractional-love model with the external force and the fractional-order system was analyzed when the parameters are fixed. Further, we also studied the relationship between the chaotic systemic dynamic and the parameters when the fractional-order system is fixed. The results show that when the parameters are fixed, the fractional-order system exhibited segmented chaotic states for the different fractional orders of the system. When fixed the fractional-order system, the system exhibited the periodic and chaotic states as parameter changes.

Let g be a strictly increasing function on $\left(a,b\right),$ having a continuous derivative g′ on $\left(a,b\right).$ For the Lebesgue integrable function $f:\left(a,b\right)\to \mathbb{C}$ , we define the k-g-left-sided fractional integral of f by $S_{k,g,a+}f\left(x\right)={\int }_a^{x}k\left(g\left(x\right)-g\left(t\right)\right)g^{\prime }\left(t\right)f\left(t\right)dt,x\in \left(a,b\right]$ and the k-g-right-sided fractional integral of f by $S_{k,g,b-}f\left(x\right)={\int }_x^{b}k\left(g\left(t\right)-g\left(x\right)\right)g^{\prime }\left(t\right)f\left(t\right)dt,x\in [a,b),$ where the kernel k is defined either on $\left(0,\infty \right)$ or on $\left[0,\infty \right)$ with complex values and integrable on any finite subinterval. In this paper we establish some Ostrowski and trapezoid type inequalities for the k-g-fractional integrals of functions of bounded variation. Applications for mid-point and trapezoid inequalities are provided as well. Some examples for a general exponential fractional integral are also given.

The fractional telegraph partial differential equation with fractional Atangana-Baleanu-Caputo (ABC) derivative is studied. Laplace method is used to find the exact solution of this equation. Stability inequalities are proved for the exact solution. Difference schemes for the implicit finite method are constructed. The implicit finite method is used to deal with modelling the fractional telegraph differential equation defined by Caputo fractional of Atangana-Baleanu (AB) derivative for different interval. Stability of difference schemes for this problem is proved by the matrix method. Numerical results with respect to the exact solution confirm the accuracy and effectiveness of the proposed method.

Nonalcoholic fatty liver disease (NAFLD) is becoming the leading cause of hepatocellular carcinoma (HCC), liver-related mortality, and liver transplantation. There is reasonable epidemiological cohort data to recommend surveillance of patients with NAFLD based upon the incidence of HCC. The American Gastroenterology Association (AGA) expert review published in 2020 recommend that NAFLD patients with cirrhosis or advanced fibrosis estimated by non-invasive tests (NITs) should consider HCC surveillance. NITs include fibrosis-4 (FIB-4) index, the enhanced liver fibrosis (ELF) test, FibroScan, and MR elastography. The recommended surveillance modality is abdominal ultrasound (US) given that it is cost effective and noninvasive with good sensitivity. However, US is limited in obese patients and those with NAFLD. In NAFLD patients with a high likelihood of having an inadequate US or if US is attempted but inadequate, CT or MRI may be utilized. The GALAD score, consisting of age, gender, AFP, lens culinaris-agglutinin-reactive fraction of AFP (AFP-L3), and protein induced by vitamin K absence or antagonist-II (PIVKA-II), can help to identify high risk of incident HCC in NAFLD patients. Innovative parameters including Mac-2 binding protein glycated isomer , type IV collagen 7S, free apoptosis inhibitor of macrophage, combination of single nucleoside polymorphisms are expected to be established. Considering a large number of NAFLD population, optimal screening tests must meet several criteria including high sensitivity, cost effectiveness and availability.

Recently, representation formulae and monotonicity properties of generalized k-Bessel functions, Wk v,c., were established and studied by SR Mondal [24]. In this paper, we pursue and investigate some of their image formulae. We then extract solutions for fractional kinetic equations, involving Wk v,c, by means of their Sumudu transforms. In the process, Important special cases are then revealed, and analyzed.

Speckle noise corrupt the major part of ultrasound image, because of which the quality deteriorate and loss of valuable information leads to false diagnosis. A large community of images like synthetic aperture radar (SAR) image, Synthetic image, and simulated ultrasound image, require despeckling at pre-processing stage for better processing. Cleaning the speckle from image and preserving the edge details is a vital task. Nowadays not only despeckling is considered as an important process but also preserving information at boundary and edges of image is also important. As most of the algorithms able to remove speckle noise but fails to preserve the details of edges. This paper covers several recent methods for removal of speckle noise along with various metrics opted for comparisons. The distinctive part of this paper is, a mathematical and parametric review has been done. Also a table is also included which summarizes the entire paper.

The Chinese drug Bombyx Batryticatus mori.L which also named as the white stiff silkworm is widely used in clinics, due to the significant antispasmodic and promotional blood circulation effects. In addition, its hypoglycemic effect is also recognized in recent years. From a pathological point of view, the enzymatic glycosylation and non-enzymatic glycation both have important roles in regulating properties of proteins and are associated with Diabetes. With the db/db mouse model, we examined the alterations of N-glycosylation of diabetic myocardium at primary stage and clarify the differences in glycosylation of myocardium before and after with 1-DNJ treatment. Hydrophilic chromatography solid phase extraction enrichment and LC-MS/MS identification was applied to profile the alternations in protein glycosylation. Meanwhile, N-glycan α1, 6-fucosylation alterations were profiled with LCA lectin blot and FITC-labelled lectin affinity histochemistry. Our results showed that AGES, hydroxyproline, CTGF and other serum indicators and fibrosis related cytokines expressional levels were reduced significantly by 1-DNJ in a dose-dependent manner. In order to verify this result, the well-known pathway of TGF-β/smad2/3 was picked out and α1, 6-core fucosylated TGFR-βⅡwas semi-quantified with western blot method. The result sustained the conclusion from LCA lectin affinity histochemistry and lectin blot analysis. The expressional level of α1, 6-fucosyltransferase mRNA was increased in the myocardium of db/db mice, however, the 1-DNJ administration did not show obvious inhibitory effect on FU8 expression. This unexpected result can be interpreted as 1-DNJ plays the roles by reducing the concentration of substrate rather than inhibiting α1，6-fucose glycosyltransferase expression. Meanwhile, 1-DNJ crude extract from BBm with some flavonoids accompany can also play the roles of anti-oxidant, and all the chemicals protect the diabetic myocardium from hyperglycemia damage commonly.

An expansion method based on time fractional Sine-Gordon equation is implemented to construct some real and complex valued exact solutions to the Korteweg-de Vries and modified Korteweg-de Vries equation in time fractional forms. Compatible fractional traveling wave transform plays a key role to be able to apply homogeneous balance technique to set the predicted solution. The relation between trigonometric and hyperbolic functions based on fractional Sine-Gordon equation allows to form the exact solutions with multiplication of powers of hyperbolic functions.

In this paper we study the class of mixed-index time fractional differential equations in which different components of the problem have different time fractional derivatives on the left hand side. We prove a theorem on the solution of the linear system of equations, which collapses to the well-known Mittag-Leffler solution in the case the indices are the same, and also generalises the solution of the so-called linear sequential class of time fractional problems. We also investigate the asymptotic stability properties of this class of problems using Laplace transforms and show how Laplace transforms can be used to write solutions as linear combinations of generalisedMittag-Leffler functions in some cases. Finally we illustrate our results with some numerical simulations.

In this paper, we established the existence of PC-mild solutions for non local fractional impulsive functional integro-differential equations with finite delay. The proofs are obtained by using the techniques of fixed point theorems, semi-group theory and generalized Bellman inequality. In this paper, we have used the distributed characteristic operators to define the mild solution of the system. Results obtained here improve and extend some known results.

The main objective of this present paper is to introduce further extension of extended Caputo fractional derivative operator and establish the extension of an extended fractional derivative of some known elementary functions. Also, we investigate the extended fractional derivative of some familiar special functions, the Mellin transforms of newly defined Caputo fractional derivative operator and generating relations for extension of extended hypergeometric functions.

In this paper, we pursue and investigate the solutions for fractional kinetic equations, involving Bessel-Struve function by means of their Sumudu transforms. In the process, one Important special case is then revealed, and analyzed. The results obtained in terms of Bessel-Struve function are rather general in nature and can easily construct various known and new fractional kinetic equations.

The present work is concerned with the study of the generalized Langevin equation and its link to the physical theories of statistical mechanics and scale relativity. It is demonstrated that the form of the coefficients of Langevin equation depend critically on the assumption of continuity of the reconstructed trajectory. This in turn demands for the fluctuations of the diffusion term to be discontinuous in time. This paper further investigates the connection between the scale-relativistic and stochastic mechanics approaches, respectively, with the study of the Burgers equation, which in this case appears as a stochastic geodesic equation. By further demanding time reversibility of the drift the Langevin equation can also describe equivalent quantum-mechanical systems in a path-wise manner. The resulting statistical description obeys the Fokker-Plank equation of the probability density of the differential system, which can be readily estimated from Monte Carlo simulations of the random paths. Based on the Fokker-Plank formalism a new derivation of the transient probability densities is presented. Finally, stochastic simulations are compared to the theoretical results.

This article deals with the conformable double Laplace transforms and their some properties with examples and also the existence Condition for the conformable double Laplace transform is studied. Finally, in order to obtain the solution of nonlinear fractional problems, we present a modified conformable double Laplace that we call conformable double Laplace decomposition methods (CDLDM). Then, we apply it to solve, Regular and singular conformable fractional coupled burgers equation illustrate the effectiveness of our method some examples are given.

We do fuzzification the concept of domination in crisp graph by using membership values of nodes, α-strong and arcs. In this paper, we introduce a new variation on the domination theme which we call vertex domination. We determine the vertex domination number γ_v for several classes of fuzzy graphs, specially complete fuzzy graph and complete bipartite fuzzy graphs. The bounds is obtained for the vertex domination number of fuzzy graphs. Also the relationship between M-strong arcs and α-strong is obtained. In fuzzy graphs, monotone decreasing property and monotone increasing property is introduced. We prove the vizing’s conjecture is monotone decreasing fuzzy graph property for vertex domination. we prove also the Grarier-Khelladi’s conjecture is monotone decreasing fuzzy graph property for it. We obtain Nordhaus-Gaddum (NG) type results for these parameters. The relationship between several classes of operations on fuzzy graphs with the vertex domination number of them is studied.

In this paper, we generalize the problems of finding simple polygons with the minimum area, maximum perimeter and maximum number of vertices so that they contain a given set of points and their angles are bounded by $\alpha+\pi$ where $\alpha$ ($0\leq\alpha\leq \pi$) is a parameter. We also consider the maximum angle of each possible simple polygon crossing a given set of points, and derive an upper bound for the minimum of these angles. The correspondence between the problems of finding simple polygons with the minimum area and maximum number of vertices is investigated from a theoretical perspective. We formulate the three generalized problems as nonlinear programming models, and then present a Genetic Algorithm to solve them. Finally, the computed solutions are evaluated on several datasets and the results are compared with those from the optimal approach.

The main objective of this present paper is to establish the extension of an extended fractional derivative operator by using an extended beta function recently defined by Parmar et al. by considering the Bessel functions in its kernel. Also, we give some results related to the newly defined fractional operator such as Mellin transform and relations to extended hypergeometric and Appell's function via generating functions.

Pulsed-field gradient (PFG) diffusion experiments can be used to measure anomalous diffusion in many polymer or biological systems. However, it is still complicated to analyze PFG anomalous diffusion, particularly the finite gradient pulse width (FGPW) effect. In practical applications, the FGPW effect may not be neglected such as in clinical diffusion magnetic resonance imaging (MRI). Here, two significantly different methods are proposed to analyze PFG anomalous diffusion: the effective phase shift diffusion equation (EPSDE) method and an observing the signal intensity at the origin method. The EPSDE method describes the phase evolution in virtual phase space, while the method to observe the signal intensity at the origin describes the magnetization evolution in real space. However, these two approaches give the same general PFG signal attenuation including FGPW effect, which can be numerically evaluated by a direct integration method. The direct integration method is fast and without overflow. It is a convenient numerical evaluation method for Mittag-Leffler function type PFG signal attenuation. The methods here provide a clear view of spin evolution under field gradient, and their results will help the analysis of PFG anomalous diffusion.

Suitable measures of grazing impacts on ground cover, that enable separation of the effects of climatic variations, are needed to inform land managers and policy makers across the arid rangelands of the Northern Territory of Australia. This work developed and tested a time-series, change-point detection method for application to time series of vegetation fractional cover derived from Landsat data to identify irregular and episodic ground-cover growth cycles. These cycles were classified to distinguish grazing impacts from that of climate variability. A measure of grazing impact was developed using a multivariate technique to quantify the rate and degree of ground cover change. The method was successful in detecting both long term (> 3 years) and short term (< 3 years) growth cycles. Growth cycle detection was assessed against rainfall surplus measures indicating a relationship with high rainfall periods. Ground cover change associated with grazing impacts was also assessed against field measurements of ground cover indicating a relationship between both field and remotely sensed ground cover. Cause and effects between grazing practices and ground cover resilience can now be explored in isolation to climatic drivers. This is important to the long term balance between ground cover utilisation and overall landscape function and resilience.

Background and objectives: The hepatoprotective activity of vitamin E and C is evident due to their ability of modulating the antioxidant pathway. In this study, we have evaluated the effects of α-tocopherol and ascorbic acid on paracetamol induced liver damage with offsetting various levels of drug treatment following an in vivo experimental protocol on Wistar albino male rats. Materials and Methods: The level of lipid peroxidation as well as histological examination of liver tissues were observed among 50 Wistar albino male rats to evaluate hepatoprotective effect of α-tocopherol and ascorbic acid on hepatocytes. The experiment was divided into 5 groups (10 rats in each group)- Basal control group (Group-I, with propylene glycol), Paracetamol treated control group (Group –II), α-tocopherol pretreated & paracetamol treated group (Group –III), Ascorbic acid pretreated & paracetamol treated group (Group –IV) and Ascorbic acid pretreated & paracetamol treated group (Group –IV). Results: The mean (± SD) Malondialdehyde (MDA) concentration were significantly reduced in α-tocopherol pretreated and paracetamol treated group (P<0.001), Ascorbic acid pretreated and paracetamol treated group (P≤0.05) and combined α-tocopherol with ascorbic acid pretreated & paracetamol treated group (P<0.001). Statistically significant differences in histological findings of rat liver were observed in paracetamol treated control group (P<0.001), ascorbic acid pretreated and paracetamol treated group (P<0.001). The serum alanine aminotransferase (ALT) level was also significantly higher in paracetamol treated group (P<0.001), α-tocopherol pretreated plus paracetamol treated group (P≤0.05) and in ascorbic acid pretreated plus paracetamol treated group (P<0.001). Conclusion: The combined pretreatment of α-tocopherol & ascorbic acid have better hepatoprotective effects than α-tocopherol or ascorbic acid alone against paracetamol induced liver damage. The decrement of free radicals produced by vitamin E could be a better hepatoprotective antioxidant than vitamin C in paracetamol induced toxicity.

The expanding use of devices emitting Pulsed Telecommunication Signals (PTS) has launched a serious debate over the possible effects of electromagnetic radiation (EMR) on living organisms. Our previous work has indicated that PTS exposure alters Amyloid Precursor Protein (APP) and alpha-synuclein (α-syn) metabolism in human cells of neural origin, providing a possible connection between exposure and neurodegeneration. This investigation aimed to reveal, in vitro in human non-neural cells (HEK293), the aftermath of the same exposure on the processing of APP and α-syn. Data presented here, indicate changes in APP metabolism, acquisition of different cellular topologies of the newly generated APP fragments, changes in monomeric α-syn accumulation and multimerization, indicating that APP and α-syn processing is possibly altered in the periphery by EMR. These effects are accompanied by a substantial increase in the levels of Reactive Oxygen Species (ROS). Further investigation is required in order to provide insights into the interaction of PTS with non-neural cells affecting the peripheral systemic functional stability. This is necessary because nowadays whole body human exposure from various EMR sources is a fact in normal life with the valid estimation that they may be increased in view of the forthcoming 5G telecommunications network implementation.

This article aims to solve the problem of robust supervision of a reverse osmosis desalination system (RO-DS) with two models, external model and bond graph model. The structure of an industrial system from the point of view of the external model operates according to several modes of operation (degraded and normal). For this external model, we cannot locate the faults since we are talking about a global operation of the system. The possible solution for the development of research was to use the multidisciplinary model called bond graph. This model by its graphic nature and using a unified language makes it possible to model the industrial system element by element from where it helps the user not only to detect the faults but also to locate them when they appear in the system. The results suggest that the use of the bond graph model for alarm 02 is a reverse osmosis error (RO1); these phenomena are readable on the bond graph model and can be quantified by equations. The equation of the model of this reverse osmosis (RO1) is found in the residual equations (r5, r6 and r7), so that these residues will be sensitive to this rupture. The possible solution during research development was to use the bond graph model to supplement information with physical knowledge and to locate faults. This article also describes the operating safety (by minimizing false alarms and non-detections as well as delays in fault detection) of the desalination system by using the bond graph model described in Linear Fractional Transformation (LFT) form for enable it to manage the robust supervision of a desalination system. An approach based on (LFT-BG) is developed to monitor tank leakage (Cu) and reverse osmosis (RO1 and RO2) faults or valve level (V1 and V2) closures and membrane clogging reverse osmosis (Rm1 and Rm2) that can occur in the reverse osmosis desalination system (RO-DS).

DGJ (Daftardar-Gejii-Jafaris) method is used to obtain numerical solution of the third order fractional differential equation. Providing the DGJ method converges, the approximate solution is a good and effective numerical result which is close to the exact solution or the exact solution. For this,the examples of the explaning the method are presented. The proposed method is implemented for the approximation solution of the third order nonlinear fractional partial differential equations. The method was shown to be unsuitable and inconsistent for an example of a nonlinear fractional partial differential equation depend on initial-boundary value conditions. The fact that these numerical results are not consistent can be explained by the fact that the method is not convergent.

Fractional differential equations have become a fundamental modelling approach for understanding and simulating the many aspects of nonlocality and spatial heterogeneity of complex materials and systems. Yet, while real-order fractional operators are nowadays widely adopted, little progress has been made in extending such operators to complex-order counterparts. In this work, we introduce new definitions for the complex-order fractional Laplacian, fully consistent with the theory of averaging of smooth functions over fractal sets, and present tailored spectral methods for their numerical treatment. The proposed complex-order operators exhibit a dual particle-wave behaviour, with solutions manifesting wave-like features depending on the magnitude of the imaginary part of the fractional order. Reaction-diffusion systems driven by the complex-order fractional Laplacian exhibit unique spatio-temporal dynamics, such as equilibrium of diffusion in random materials by interference of scattered waves, conduction block and highly fractionated propagation, or the generation of completely novel Turing patterns. Taken together, our results support that the proposed complex-order operators hold unparalleled capabilities to advance the description of multiscale transport phenomena in physical and biological processes highly influenced by the heterogeneity of complex media.

In the present research, we will develop some integral inequalities of Hermite Hadamard type for differentiable η- convex function. Moreover, our results include several new and known results as special cases.

This paper deals with the analysis of trends in road accidents on major highways in Brazil. Using updated time series techniques, our results indicate that a low degree of long memory was detected in the series with shocks having transitory effects over time. We further find that the number of accidents taking place in Brazil has been reducing over time, though in the presence of negative shocks, the recovery is not going to be immediate due to the long memory nature of the data. Despite the absence of relevant investment relating to infrastructure expansion, it is worth mentioning the consolidation of a nationwide tolled road system in Brazil involving concessions to private administrators, alongside more severe traffic laws that can impose limitations on driving licences.

Rough volatility models are popularized by \cite{gatheral2018volatility}, where they have shown that the empirical volatility in the financial market is extremely consistent with rough volatility. Fractional Riccati equation as a part of computation for the characteristic function of rough Heston model is not known in explicit form as of now and therefore, we must rely on numerical methods to obtain a solution. In this paper, we give a short introduction to option pricing theory and an overview of the current advancements on the rough Heston model.

Creating self-learning algorithms, developing deep neural networks and improving other methods that "learn" for various areas of human activity is the main goal of the theory of machine learning. It helps to replace the human with a machine, aiming to increase the quality of production. The theory of artificial neural networks, which already have replaced the humans in problems of detection of moving objects, recognition of images or sounds, time series prediction, big data analysis and numerical methods remains the most dispersed branch of the theory of machine learning. Certainly, for each area of human activity it is necessary to select appropriate neural network architectures, methods of data processing and some novel tools from applied mathematics. But the universal problem for all these neural networks with specific data is the achieving the highest accuracy in short time. Such problem can be resolved by increasing sizes of architectures and improving data preprocessing, where the accuracy rises with the training time. But there is a possibility to increase the accuracy without time growing, applying optimization methods. In this survey we demonstrate existing optimization algorithms of all types, which can be used in neural networks. There are presented modifications of basic optimization algorithms, such as stochastic gradient descent, adaptive moment estimation, Newton and quasi-Newton optimization methods. But the most recent optimization algorithms are related to information geometry, for Fisher-Rao and Bregman metrics. This approach in optimization extended the theory of classic neural networks to quantum and complex-valued neural networks, due to geometric and probabilistic tools. There are provided applications of all introduced optimization algorithms, what delighted many kinds of neural networks, which can be improved by including any advanced approaches in minimization of the loss function. Afterwards, we demonstrated ways of developing optimization algorithms in further researches, engaging neural networks with progressive architectures. Classical gradient based optimizers can be replaced by fractional order, bilevel and, even, gradient free optimization methods. There is a possibility to add such analogues in graph, spiking, complex-valued, quantum and wavelet neural networks. Besides the usual problems of image recognition, time series prediction, object detection, there are many are other tasks for modern theory of machine learning, such as solving problem of quantum computations, partial differential and integro-differential equations, stochastic processes and Brownian motion, making decisions and computer algebra.

The optical conductivity in the charge order phase is calculated in the extended Hubbard model describing an organic Dirac electron system α-(BEDT-TTF)₂I₃ using the mean field theory and the Nakano-Kubo formula. A peak structure due to interband excitation being characteristic in two-dimensional Dirac electron system is found above the charge order gap. It is shown that the peak structure originates from the Van Hove singularities of the conduction and valence bands, where those singularities are located at a saddle point between two Dirac cones in momentum space. The frequency of the peak structure exhibits drastic change in the vicinity of the charge order transition.

In the search of potent α-amylase inhibitors, we have synthesized seventeen derivatives of 2-mercaptobenzimidazole bearing sulfonamide (1-17) and evaluated for their α-amylase inhibitory potential. All compounds display a variable degree of α-amylase activity having IC50 values ranging between 0.90 ± 0.05 to 11.20 ± 0.30 µM when compared with the standard drug acarbose having IC50 value 1.70 ± 0.10 µM. Compound 1, 2, 11, 12 and 14 having IC50 values 1.40 ± 0.10, 1.30 ± 0.05, 0.90 ± 0.05, 1.60 ± 0.05 and 1.60 ± 0.10 µM respectively were found many folds better than the standard drug acarbose. The remaining analogs showed good inhibitory potentials. All the synthesized compounds were characterized by HREI-MS, 1H and 13C-NMR. Structure activity relationship (SAR) has been recognized for all newly synthesized analogs. Through molecular docking study, binding mode of active analogs with α-amylase enzyme was confirmed.

Neuronal lesions in Parkinson’s disease (PD) are commonly associated with α-synuclein (α-Syn)-induced cell damage that are present both in the central and peripheral nervous systems of patients, with the enteric nervous system also being especially vulnerable. Here we bring together evidence that the development and presence of PD depends on specific sets of interlinking factors that include neuro-inflammation, systemic inflammation, α-Syn-induced cell damage, vascular dysfunction, iron dysregulation, gut and periodontal dysbiosis. We argue that there is significant evidence that bacterial inflammagens fuel this systemic inflammation, and might be central to the development of PD. We also discuss the processes whereby lipopolysaccharides may be involved in causing nucleation of proteins, including of α-Syn. Lastly, we review evidence that pre-and probiotics, as well as antibiotics and faecal transplant treatment might be valuable treatments in PD. A most important consideration, however, is that these therapeutic options need to be validated and tested in randomized controlled clinical trials. However, targeting underlying mechanisms of PD, including gut dysbiosis and iron toxicity, have potentially opened up possibilities of a wide variety of novel treatments which may relieve the characteristic non-motor deficits of PD, and may even slow the progression and/or accompanying gut-related conditions of the disease.

This paper uses the $\exp(-\Phi(\xi))$-Expansion method to investigate solitons to the M-fractional nonlinear Schrödingers equation with cubic nonlinearity. The results obtained are dark solitons, trigonometric function solutions, hyperbolic solutions and rational solutions. Thus, the constraint relations between the model coefficients and the traveling wave frequency coefficient for the existence of solitons solutions are also derived.

Reducing carbohydrates digestion by having low glycaemic index (GI) foods has been linked to weight loss. Inhibiting related enzymes is an alternative way to decrease carbohydrate digestion. RCM-107, an eight-herb formula that is modified from the RCM-104, has indicated significant weight-loss action in clinical trials. However, no research has been conducted to study its effect on the activity of porcine pancreatic alpha-amylase (PPA), which is involved in carbohydrate absorption. In this paper, we used fluorescence PPA inhibition assay to investigate the inhibitory effects of RCM-107 and the individual herbs present in this herbal mixture on amylase activity. Subsequently, molecular docking predicted the key active compounds that may be responsible for the enzyme inhibition. According to our results, both the RCM-107 formula and several individual herbs displayed α-amylase inhibitory effects. Also, marginal synergistic effects of RCM-107 were also detected. In addition, alisol B, (-)-epigallocatechin-3-gallate (EGCG) and plantagoside have been predicted as the key active compounds that may be responsible for the α-amylase inhibition effect of RCM-107 according to inter-residue contact analysis. Finally, Glu233, Gln63, His305, Asp300 and Tyr151 are predicted to be markers of important areas with which potential amylase inhibitors would interact.

Fractional calculus gained a lot of attention in the last couple of years. Researchers discovered that processes in various fields follow rather fractional dynamics than ordinary integer-ordered dynamics, meaning the corresponding differential equations feature non-integer valued derivatives. There are several arguments for why this is the case, one of them being that fractional derivatives’ inherit spatiotemporal memory and/or the ability to express complex naturally occurring phenomena. Another popular topic nowadays is machine learning, i.e., learning behavior and patterns from historical data. In our ever-changing world with ever-increasing amounts of data, machine learning is a powerful tool for data analysis, problem-solving, modeling, and prediction. It further provides many insights and discoveries in various scientific disciplines. As these two modern-day topics provide a lot of potential for combined approaches to describe complex dynamics, this article reviews combined approaches of fractional derivatives and machine learning from the past, puts them into context, and thus provides a list of possible combined approaches and the corresponding techniques. Note, however, that this article does not deal with neural networks, as there already is profound literature on neural networks and fractional calculus. We sorted past combined approaches from the literature into three categories, i.e., preprocessing, machine learning &amp; fractional dynamics, and optimization. The contributions of fractional derivatives to machine learning are manifold as they provide powerful preprocessing and feature augmentation techniques, can improve physically informed machine learning, and are capable of improving hyperparameter optimization. Thus, this article serves to motivate researchers dealing with data-based problems, to be specific machine learning practitioners, to adopt new tools and enhance their existing approaches.

Recently, the conformable derivative and its properties have been introduced. In this paper, we propose and prove some new results on conformable Boundary Value Problems. First, we introduce a conformable version of classical Sturm´s separation, and comparison theorems. For a conformable Sturm-Liouville problem, Green's function is constructed, and its properties are also studied. In addition, we propose the applicability of the Green´s Function in solving conformable inhomogeneous linear differential equations with homogeneous boundary conditions, whose associated homogeneous boundary value problem has only trivial solution. Finally, we prove the generalized Hyers-Ulam stability of the conformable inhomogeneous boundary value problem.

This paper describes a proposed fractional filter-based multi-scale underwater and hazy image enhancement algorithm. The proposed system combines a modified global contrast operator with fractional order-based multi-scale filters used to generate several images, which are fused based on entropy and standard deviation. The multi-scale-global enhancement technique enables fully adaptive and controlled colour correction and contrast enhancement without over exposure of highlights when processing hazy and underwater images. This in addition to illumination/reflectance estimation coupled with global and local contrast enhancement. The proposed algorithm is also compared with the most recent available state-of-the-art multi-scale fusion de-hazing algorithm. Experimental comparisons indicate that the proposed approach yields better edge and contrast enhancement results without halo effect, colour degradation and is faster and more adaptive than all other algorithms from the literature.

The abnormal assembly of α-synuclein (α-Syn) is an initial step in the formation of Lewy bodies in the brain, which finally causes the neuronal death, being considered as a pathological hallmark in Parkinson’s disease (PD). Certain food bioactives or their metabolites at very low concentrations can trespass the blood brain barrier (BBB) that might, thereafter, act simultaneously. The aim of this work was to evaluate the inhibitory and destabilising capacities on α-Syn kinetics and the neuroprotective effects of three well-known bioactive compounds able to cross the BBB and present in foods; melatonin (MEL), protocatechuic acid (PCA) and hydroxytyrosol (HT), and their combinations. For this purpose, different in vitro techniques (Thioflavin T (ThT), Transmission Electronic Microscopy (TEM), electrophoresis and MTT assay) were used. All tested compounds and their combinations were able to abolish the toxicity induced by α-Syn. In addition, the combination of PCA (100 µM) +HT (100 µM) showed the highest inhibitory effect against α-Syn fibril formation and destabilises α-Syn fibrils (88 and 62%, respectively). This is the first time that MEL, PCA and HT prove a joint effect against α-Syn aggregation and toxicity when they are tested together.

Plukenetia volubilis is an underutilized oilseed crop native to the Amazon basin, where it has been utilised by humans since Incan times. The large seeds contain approx. 45–50 % lipid, of which approx. 35.2–50.8 % is α-linolenic acid (C18:3 n-3, ω-3) and approx. 33.4–41.0 % is linoleic acid (C18:2 n-6, ω-6), the two essential fatty acids required by humans. The seeds also contain 22–30 % protein and have antioxidant properties. Due to its excellent nutritional composition and good agronomic properties, it has attracted increasing attention in recent years, and cultivation is expanding. When considering current global challenges, a reformation of our food systems is imperative in order to ensure food security, mitigation of climate change, and alleviation of malnutrition. For this purpose, underutilized crops may be essential tools, which can provide agricultural hardiness and reduced need for external inputs, climate resilience, diet diversification, and improved income opportunities for smallholders. Plukenetia volubilis is a promising up and coming crop in this regard and has considerable potential for further domestication; it has an exceptional oil composition, good sensory acceptability, is well suited for cultivation, and has numerous potential applications in, e.g. gastronomy, medicine, and cosmetics.

The use of salt rock for underground radioactive waste disposal facilities requires a comprehensive analysis of creep-damage process in salt rock. A computer-controlled creep setup is employed to carry out a creep test of salt rock lasted as long as 359 days under a constant uniaxial stress. The AE space-time evolution and energy releasing characteristics during creep test are studied in the meantime. A new creep-damage model is proposed on the basis of fractional derivative by combining the AE statistical regularity. It indicates that the AE data in non-decay creep process of salt rock can be divided into three stages. Furthermore, the parameters of new creep-damage model are determined by Quasi-Newton method. The fitting analysis suggests that the creep-damage model based on fractional derivative in this paper provides a precise description of full creep regions in salt rock.

To elucidate the chemical compositions of the sugar fall maple leaves, the methanol extracts were firstly fractionated by ethyl acetate and n-butanol respectively. The phenolic acids-rich fractions (ethyl acetate extracts) were further purified by various chromatographic columns including XAD macroporous resin, Sephadex LH-20, ODS and semi-preparative HPLC to yield the compounds. The isolated compounds were characterized by ¹H-Nuclear Magnetic Resonance (¹H-NMR), ¹³C-NMR, and high resolution electrospray ionisation mass spectral (HR-ESI-MS) spectroscopy. Twenty eight phenolics including fourteen flavonoids (1-14), five quinic acid derivatives (15-19), five galloyl tannins (20-24) and four other phenolic acids (25-28) were isolated and their structures were identified. The isolated compounds were evaluated for their antioxidant and α-glucosidase inhibitory activities. All of the phenolics constituents showed DPPH scavenging antioxidant activities. While, glycosides of quercetin and myricetin, galloyl tannins were showed promising α-glucosidase inhibitory activity. All of the compounds except 4, 11, 12 and 28 were isolated from sugar maple for the first time. Moreover, Compounds 9, 10, 14, 20, 21, 23, 25 and 26 were isolated from the Acer species for the first time.

In this study, we actually want to explore the time-fractional Phi-four equation via two methods, the exp _a function method and the hyperbolic function method. We transform a fractional order dierential equation into an ordinary differential equation using a wave transformation and the fractional derivative in conformable form. Then, the resulting equation has successfully been explored for new explicit exact solutions. The procured solutions are simply showed the effectiveness and plainness of the projected methods.

To determine the mode of action of the effects of phytoalexins in soybeans, we analyzed enzyme inhibition kinetics using Michaelis–Menten plots and the Lineweaver–Burk plots. The results showed that glyceollin showed competitive inhibition, genistein showed noncompetitive, daidzein was uncompetitive, and luteolin showed a mixed mode of action. The Ki values were determined using a Dixon plot as: glyceollin, 18.99; genistein, 15.42; luteolin, 16.81; and daidzein, 9.99 μM, respectively. Furthermore, potential synergistic effects between glyceollin and the three designated polyphenols were investigated. A combination of glyceollin and luteolin (the ratio of 3:7 of glyceollin and luteolin) had synergistic effects on α-glucosidase inhibition according to combination index (CI)-isobologram equation. Collectively, these results showed that a combination of glyceollin and luteolin has the potential to inhibit α-glucosidase activity via a synergistic mode of action.

In the present research, we will develop some integral inequalities of Hermite Hadamard type for differentiable η-convex function. Moreover, our results include several new and known results as special cases.

The central limit theorem (CLT) and its generalization to stable distributions have been widely described in literature. However, many variations of the theorem have been defined and often their applicability in practical situations is not straightforward. In particular, the applicability of the CLT is essential for a derivation of heterogeneous ensemble of Brownian particles (HEBP). Here, we analyze the role of the CLT within the HEBP approach in more detail and derive the conditions under which the existing theorems are valid.

The differential transformation, an approach based on Taylor's theorem, is proposed as convenient for finding exact or approximate solution to the initial value problem with multiple Caputo fractional derivatives of generally non-commensurate orders. The multi-term differential equation is first transformed into a multi-order system and then into a system of recurrence relations for coefficients of formal fractional power series. The order of the fractional power series is discussed in relation to orders of derivatives appearing in the original equation. Application of the algorithm to an initial value problem results in a reliable and expected outcome.

In this study we used a non-autonomous Chua’s Circuit, and the fractional Lorenz chaos system together with a detection method from Extension theory to analyze the voltage signals. The measured bearing signals by acceleration sensor were introduced into the master and slave systems through a Chua’s Circuit. In a chaotic system minor differences can cause significant changes that generate dynamic errors, and extension matter-element models can be used to judge the bearing conditions. Extension theory can be used to establish classical and sectional domains using the dynamic errors of the fault conditions. The results obtained were compared with those from Discrete Fourier Transform analysis, Wavelet analysis and an integer order chaos system. The diagnostic ratio showed the fractional order master and slave chaos system calculations. The results show that the method presented in this paper is very suitable for monitoring the operational state of ball bearing system to be superior to the other methods. The diagnosis ratio was better and there were other significant advantages such as low cost and few.

Hematite (-Fe2O3) with uniform hexagonal flakes morphology has been successfully synthesized using a combination of gelatin as natural template with F127 via hydrothermal method. The resulting hematite was investigated as adsorbent and photocatalyst for removal of ibuprofen as pharmaceutical waste. Hexagonal flake-like hematite was obtained following calcination at 500 oC with the average size was measured at 1-3 µm. Increasing the calcination temperature to 700 oC transformed the uniform hexagonal structure into cubic shape morphology. Hematite also showed high thermal stability with increasing the calcination temperatures, however, the surface area was reduced from 47 m2/g to 9 m2/g. FTIR analysis further confirmed the formation Fe-O-Fe bonds, and the main constituent elements of Fe and O were observed in EDX analysis for all samples. Fe2O3-G samples have an average adsorption capacity of 55-25.5 mg/g at 12-22% of removal efficiency when used as adsorbent for ibuprofen. The adsorption capacity was reduced with increasing the calcination temperatures due to the reduction of available surface area of the hexagonal flakes when transformed into cube. Photocatalytic degradation of ibuprofen using hematite flakes achieved 50% of removal efficiency meanwhile combination of adsorption and photocatalytic degradation further removed 80% of ibuprofen in water/hexane mixtures.

The Landau damping effect was observed in collisionless plasma, as a microscopic resonant mechanism between electromagnetic radiation and the collective modes. In this paper we demonstrate the occurrence of the Landau damping at macroscopic scale in the interaction between water waves and anharmonic lattice of magnetic buoys. By coupling the Navier-Stokes equations for incompressible fluid with the nonlinear dynamics of an anharmonic magnetic lattice we obtain a resonant transfer of momentum and energy between the two systems. The velocity of the flow is obtained in the Stokes approximation with Basset type of drag force. The dynamics of the buoys is calculated in the surfactant approximation for a specific frequency, then we use Fourier analysis to obtain the general time variable interaction. After involving an integral Dirichlet transform we obtain the time dependent expression of the drag force, the interaction waves-lattice with a new term in the form of a Caputo fractional derivative. We compare the results of the model with experiments performed in a wave tank with free floating magnetic buoys under the action of small amplitude gravitational waves. This configuration can be applied in studies for the attenuation with resonant damping of rogue waves, storms or tsunamis.

In this article, we consider the three dimensional $\alpha$-fractional nonlinear delay differential system of the form \begin{align*} D^{\alpha}\left(u(t)\right)&=p(t)g\left(v(\sigma(t))\right),\\D^{\alpha}\left(v(t)\right)&=-q(t)h\left(w(t))\right),\\D^{\alpha}\left(w(t)\right)&=r(t)f\left(u(\tau(t))\right),~ t \geq t_0, \end{align*} where $0 < \alpha \leq 1$, $D^{\alpha}$ denotes the Katugampola fractional derivative of order $\alpha$. We have established some new oscillation criteria of solutions of differential system by using generalized Riccati transformation and inequality technique. The obtained results are illustrated with suitable examples.

Exact and soliton type solutions have great importance in propagation of surface waves, fluid dynamics, optics, and many other elds of nonlinear sciences. In this study, the explicit and exact soliton type solutions for two space-time fractional Equal- Width (FEW) equations with conformable derivative are procured via the hyperbolic function approach. The wave type solutions are represented in some hyperbolic and trigonometric functions.

In this paper, we compare the performance between systems of ordinary and (Caputo) fractional differential equations depicting the susceptible-exposed-infectious-recovered (SEIR) models of diseases. In order to understand the origins of both approaches as mean-field approximations of integer and fractional stochastic processes, we introduce the fractional differential equations as approximations of some type of fractional nonlinear birth--death processes. Then, we examine validity of the two approaches against empirical courses of epidemics; we fit both of them to case counts of three measles epidemics that occurred during the pre-vaccination era in three different locations. While FDEs appear more flexible in fitting empirical data, our ODEs offered better fits to two out of three data sets. Important differences in transient dynamics between these modeling approaches are discussed.

We shall discuss three methods of inverse Laplace transforms. A Sinc-Thiele approxi- mation, a pure Sinc, and a Sinc-Gaussian based method. The two last Sinc related methods are exact methods of inverse Laplace transforms which allow us a numerical approximation using Sinc methods. The inverse Laplace transform converges exponentially and does not use Bromwich contours for computations. We apply the three methods to Mittag-Leffler functions incorporating one, two, and three parameters. The three parameter Mittag-Leffler function represents Prabhakar’s function. The exact Sinc methods are used to solve fractional differential equations of constant and variable differentiation order.

This paper deals with the study of the existence and non existence of solutions of a three parameter's family of nonlinear fractional differential equation with mixed-integral boundary value conditions. We consider the $\alpha$-Riemann-Liouville fractional derivative, with $\alpha \in (1,2]$. In order to deduce the existence and non existence results, we first study the linear equation, by deducing the main properties of the related Green's functions. We obtain the optimal set of parameters where the Green's function has constant sign. After that, by means of the index theory, the nonlinear boundary value problem is studied. Some examples, at the end of the paper, are showed to illustrate the applicability of the obtained results.

The aim of this study was to assess if the ovine articular cartilage serine proteinase inhibitors (SPIs) were related to the Kunitz inter-α-trypsin inhibitor (ITI) family. Ovine articular cartilage was finely diced and extracted in 6M urea and SPIs isolated by sequential anion exchange, HA affinity and Sephadex G100 gel permeation chromatography. Selected samples were also subjected to chymotrypsin and concanavalin-A affinity chromatography. Eluant fractions from these isolation steps were monitored for protein and trypsin inhibitory activity and pooled fractions assessed by affinity blotting using biotinylated trypsin to detect active SPIs and by Western blotting using antibodies to α1-microglobulin, bikunin, TSG-6 and 2-B-6 (+) CS stub epitope generated by chondroitinase-ABC digestion. This identified 2-B-6 (+) positive 220-250,120, 58 and 36 kDa SPIs. The 58 kDa SPI contained α1-microglobulin, bikunin and chondroitin-4-sulphate stub epitope consistent with its identity as the α1-microglobulin-bikunin (AMBP) precursor and was also isolated by concanavalin-A lectin affinity chromatography indicating it had N-glycosylation. Kunitz protease inhibitor (KPI) species of 36, 26, 12 and 6 kDa could be autolytically generated by prolonged storage of the aforementioned 120 and 58 kDa SPIs; chymotrypsin affinity chromatography also generated the 6kDa SPI. KPI domain 1 and 2 SPIs were separated by concanavalin lectin affinity chromatography, domain 1 displayed affinity for this lectin indicating it had N-glycosylation. KPI 1 and 2 both displayed potent inhibitory activity towards trypsin, chymotrypsin, kallikrein, leucocyte elastase and cathepsin G. Localisation of versican, lubricin and HA in the surface regions of articular cartilage represented probable binding sites for the ITI SPs with likely importance in the preservation of joint function.

To obtain drop-in fuel properties from non-feed biomass, we herein report the catalytic hydrotreatment of microalgae biocrude, produced from hydrothermal liquefaction (HTL) of Spirulina. Our contribution focuses on the effect of temperature, initial H2 pressure, and residence time on the removal of heteroatoms (O and N). In contrast to common hydrotreating experimental protocols at batch scale, we devised a set of two-level factorial experiments and studied the most influential parameters affecting the removal of heteroatoms. It was found that up to 350 °C, the degree of deoxygenation (de-O) is mainly driven by temperature, whereas the degree of denitrogenation (de-N) also relies on initial H2 pressure and temperature-pressure interaction.Based on this, complete deoxygenation was obtained at mild operating conditions (350 °C), reaching a concurrent 47 % denitrogenation. Moreover, three optimized experiments are reported with 100 % removal of oxygen. In addition, the analysis by GC-MS and Sim-Dis gives insight to the fuel quality. The distribution of heteroatom N in lower (<340 °C) and higher (>340 °C) fractional cuts is studied by a fractional distillation unit following ASTM D-1160. Final results show that 63-68 % of nitrogen is concentrated in higher fractional cuts.

Power series expansions are useful in approximation theory and mathematical physics. The manuscript presents several types of fractional Taylor expansions of sufficiently smooth functions. This is achieved by employing an incremental regularization procedure to the computation of the derivative. The series are constructed in an algorithmic way, which allows for its implementation in computer algebra systems.

Our aim in this paper is to establish some new Hermite-Hadamard- Mercer type integral inequalities by utilizing the fractional proportional-integral operators.For this purpose, Hermite-Hadamard-Mercer inequalities for di¤er- antiable mappings whose derivatives in absolute value are convex.

The Sanshui Basin (SSB) is located at the northern continental margin of the South China Sea and characterized by a continental rift basin. The bimodal volcanic rocks in SSB record the early Cenozoic magmatic activity in the South China Block, on the magmatic evolution process of bimodal volcanic rocks are poorly understood. Clinopyroxenes in bimodal volcanic rocks in the SSB provide an opportunity to investigate the magma process during magma ascent. We classified nine types of clinopyroxene phenocrysts according to the different compositions and textures types in cogenetic basalt-trachyandesite-comenditic trachyte, the composition of unzoned clinopyroxene have an evolution sequence of diopside- hedenbergite- aegirine with the decrease of Mg#, and the trace element contents of unzoned clinopyroxenes also increase systematically during magma evolution, indicating that the genesis of clinopyroxene dominated by fractional crystallization in a closed magma system; however, the clinopyroxenes with reverse zoning and multiple zoning record the process of magma mixing and recharge indicating an open magma system. Whilst fractional crystallization is the dominated process, magma mixing, recharge, and crystal settling complicate magma evolution. Thermobarometric calculations show that clinopyroxene phenocrysts in bimodal volcanic rocks of SSB are distributed in the whole crust during magma ascent. We have established a magma plumbing system, which provides a new constrain for the complex magmatic evolution history in the SSB by detailed mineral-scale analysis.

Substrate binding proteins (SBP) bind to specific ligands in the periplasmic region and bind to membrane proteins to participate in transport or signal transduction. Typical SBPs consist of two α/β domains and recognize the substrate by hinge motion between two domains. Conversely, short length Rhodothermus marinus SBP (named as RmSBP) exists around the methyl-accepting chemotaxis protein. We previously determined the crystal structure of RmSBP consisting of a single α/β domain, but the substrate recognition mechanism is still unclear. To better understand the short length RmSBP, we performed comparative structure analysis, computational substrate docking, and X-ray crystallographic study. RmSBP shares a high level of similarity in α/β domain with other SBP proteins, but it has a distinct topology in the C-term region. The substrate binding model suggested that conformational change in the peripheral region of RmSBP was required to recognize the substrate. We determined the crystal structures of RmSBP at pH 5.5, 6.0, and 7.5. RmSBP showed structural flexibility of the β1-α2 loop, β5-β6 loop, and extended C-term domain based on the electron density map and temperature B-factor analysis. These results provide information that will further the understanding of the function of the short length SBP.

For the first time, We do fuzzification the concept of domination in crisp graph on a generalization of fuzzy graph by using membership values of vertices, α-strong edges and edges. In this paper, we introduce the first variation on the domination theme which we call vertex domination. We determine the vertex domination number γ_v for several classes of t-norm fuzzy graphs which include complete t-norm fuzzy graph, complete bipartite t-norm fuzzy graph, star t-norm fuzzy graph and empty t-norm fuzzy graph. The relationship between effective edges and α-strong edges is obtained. Finally, we discuss about vertex dominating set of a fuzzy tree with respect to a t-norm ⨂ by using the bridges and α-strong edges equivalence.

The seeds of cultivated peanut, Arachis hypogaea, are an agronomically important crop produced for human nutrition, oilseed and feed stock. Peanut seed is the single most expensive variable input cost and thus producers require seed with excellence performance in terms of germination efficiency. During the maturation process, triglycerides are stored in oil bodies as an energy resource during germination and seedling development. The stability of oil body membranes is essential for nutrient mobilization during germination. This study focused on evaluating the phytosterol composition in seed components including the kernel, embryo (heart), and seed coat or skin. Samples of different maturity classes were analyzed for macronutrient and phytosterol content. The three most abundant phytosterols, β-sitosterol, campesterol, and stigmasterol, comprised 82.29%, 86.39%, and 94.25% of seed hearts, kernels, and seed coats, respectively. Stigmasterol concentration was highest in the seed kernel providing an excellent source of this sterol known to have beneficial effects on human health. Peanut hearts contained the highest concentration of sterols by mass potentially providing protection and resources for the developing seedling. The amount of α-tocopherol increases in peanut hearts during the maturation process providing protection from temperature stress and stability required for seedling vigor. These results suggest that phytosterols may play a significant role in the performance of seeds and provides a possible explanation for the poor germination efficiency of immature seeds.

In this study, we consider delta fractional Sturm--Liouville (DFSL) initial value problems in the sense of delta Caputo and delta Riemann-Liouville (R--L) operators. One of the properties of delta fractional difference operators which makes it different from nabla counterpart is to shift its domain. This feature makes it more complex than the nabla fractional operator. We obtain sum representation of solutions for DFSL initial value problems with the help of $\mathcal{Z}-$ transformation. Moreover, we get analytical solutions of homogeneous DFSL problem within Riemann-Liouville (R--L) and Caputo sense, discrete Sturm--Liouville (DSL) problem, continuous fractional Sturm--Liouville (FSL) problem in the sense of R--L and Caputo operators, and continuous Sturm--Liouville (SL) differential problem. From this point of view, we compare all the solutions with each other. Consequently, we show that all results for these four eigenvalue problems are compatible with each other and approach to each other while the orders tends to one, i.e. $\Delta^{\mu }\left( \Delta x\left( t-\mu \right) \right)\cong D_{0^{+}}^{\mu }\left( x^{\prime }\left( t\right) \right)\cong \Delta^2x(n-1) \cong x^{\prime \prime }\left( t\right) =\lambda x\left( t\right),\ \mu\rightarrow1 $ . We support our results comparatively by tables and simulations in detail.

In this article, we establish the existence of and then compare smallest eigenvalues for nabla fractional boundary value problems involving a fractional difference boundary condition, using the theory of u0-positive operators with respect to a cone in a Banach space.

We consider a model of predator-prey interaction at fractional-order where the predation obeys the ratio-dependent functional response and the prey is linearly harvested. For the proposed model, we show the existence, uniqueness, non-negativity as well as the boundedness of the solutions. Conditions for the existence of all possible equilibrium points and their stability criteria, both locally and globally, are also investigated. The local stability conditions are derived using the Magtinon's theorem, while the global stability is proven by formulating an appropriate Lyapunov function. The occurance of Hopf bifurcation around the interior point is also shown analytically. At the end, we implement the Predictor-Corrector scheme to perform some numerical simulations.