Planar Geometry Research Articles

Synthesis and characterization of tannic acid–copper complex: A promising anticholinesterase drug

This study aimed to present an optimized synthetic pathway for a complex formed between copper (II) metal ion and tannic acid (TA), which have a variety of pharmacological properties. The study also focused on characterizing this metallodrug, carrying out in vitro antioxidant and acetylcholinesterase enzyme (AChE) inhibition assays, and performing in vivo toxicity assay against zebrafish (Danio rerio), thus expanding the area of study involving syntheses of metal complexes to act as therapeutic agents. Through various characterization techniques, including UV–Vis Absorption Spectroscopy, Fourier Transform Infrared Spectroscopy (FTIR), Thermal Analysis (TG and DTG), 1H Nuclear Magnetic Resonance (1H NMR), Inductively Coupled Plasma with Optical Emission Spectrometry (ICP-OES), and Electron Paramagnetic Resonance (EPR), it was confirmed that the metal ion is coordinated to the ligand, exhibiting distorted planar square geometry with mononuclear copper (3.10 ± 0.10 % copper by ICP-OES). In vitro tests demonstrated that the TA–Cu complex presents antioxidant activity against DPPH (IC50 = 2.26 ± 0.01 µg mL−1) and ABTS (IC50 = 1.91 ± 0.07 µg mL−1) radical scavenging assays. These results were more promising than those obtained for the TA (4.25 ± 0.03 µg mL−1 and 3.37 ± 0.03 µg mL−1, respectively). In the in vitro inhibition of AChE assay, the TA–Cu complex (4.07 ± 0.04 µg mL−1) presented a lower IC50 value than TA (5.80 ± 0.09 µg mL−1), indicating that coordination to the metal center Cu (II) was able to improve the anticholinesterase activity of the free ligand. Furthermore, the TA–Cu complex did not show toxicity in the in vivo test with adult zebrafish for 96 h at the tested doses of 4–40 mg kg−1, with LD50 >40 mg kg−1. Thus, it is estimated that the TA–Cu complex is a metallodrug with anticholinesterase potential, representing a promising strategy for conducting future pre-clinical studies in models of Alzheimer’s disease.

Using fractal voids to understand Mode I compressive fracture in brittle materials – A two-dimensional analysis

Unlike cases of direct or indirect tensile loading, analysis of Mode I fracture in cases of compressive loading must necessarily consider two-dimensional flaw geometries. The consequent analytical complications have resulted in little theoretical evolution towards understanding Mode I fracture in compressive loading. Researchers have recently observed that the linear elastic (LE) stress fields surrounding two-dimensional flaws in compression appear to have indicative value regarding Mode I fracture, but no rational framework has yet been proposed for their interpretation. Herein it is demonstrated that by idealizing void surfaces as fractal, and using what will be called the “small flaw assumption,” LE stress fields can be interpreted to obtain significant insight regarding brittle Mode I compressive fracture. Using fractal voids, a rational and consistent explanation for the satisfaction of the stress and energy criteria at the surface of two-dimensional flaws is presented. The discussion resolves many typical theoretical issues in the literature in a manner consistent with fundamental Griffith theory, and accounts for size and shape effects. The framework further enables the computationally efficient prediction of peak KI values associated with the propagation of line cracks from two-dimensional flaws from a brittle medium subject to macroscopic uniaxial compression via LE stress fields.

Synthesis, Characterization, Investigation of DNA Interactions and Biological Evaluation of Co(II), Ni(II), Cu(II) and Zn(II) Complexes with Newly Synthesized 2-methoxy 5-trifluoromethyl benzenamine Schiff Base.

The biologically active and thermally stable bivalent Co(II), Ni(II), Cu(II), and Zn(II) complexes (C1, C2, C3, and C4) of novel Schiff base ligand [(5-trifluoromethyl-2-methoxyphenylamino)methyl)-4,6-diiodophenol (L)] have been synthesized. The structural analysis of these complexes have been carried out by elemental analysis, 1H-NMR, FTIR, ESI mass, UV-visible, ESR, TGA techniques and magnetic measurements. The obtained results were confirmed as square planar geometry for Ni(II) and Cu(II) complexes, whereas octahedral geometry for Co(II) and Zn(II) complexes. The geometry optimized structures were developed by employing CHEM 3D software. The DNA binding interaction studies such as UV-vis absorption, viscosity, and fluorescence studies have been confirmed that the mode of binding of complexes with DNA is an intercalative binding. The DNA cleavage studies revealed that all the complexes are found to be potent to cleave the DNA into Form I & II. The in-vitro pathological studies of all the complexes against various microbial strains (Gram + and Gram -), revealed that Cu(II) complexes are more potent compared to other complexes and Schiff base. The anti diabetic activity studies revealed that the Cu(II) complex exhibited slightly higher activity than Co(II), Ni(II), and Zn(II) complexes. The results of antioxidant activity by DPPH method, suggested that the Cu(II) complex has higher activity and comparable with the standard compounds.

Substituents' Effect on the Photophysics of Trinuclear Copper(I) and Silver(I) Pyrazolate-Phosphine Cages.

A series of structurally similar trinuclear macrocyclic copper(I) and silver(I) pyrazolate complexes bearing various short-bite diphosphine R2PCH(R')PR2 ligands are reported. Upon diphosphine coordination, the planar geometry of the initial complexes undergoes bending along the line between two metal atoms coordinated to the phosphorus moieties. The complexes based on dcpm ligands (R = cyclohexyl, R' = H, Ph) do not exhibit dynamic behavior in solution at room temperature on the 31P NMR time scale as it was previously observed for similar trinuclear copper complexes bearing the dppm (R = Ph, R' = H) scaffold. All copper(I) complexes exhibit thermally activated delayed fluorescence (TADF) behavior in the solid state. Importantly, the use of aliphatic substituents on the phosphorus atoms instead of aromatic ones leads to an almost double increase in the quantum efficiency (ΦPL) of photoluminescence by eliminating nonradiative decay from the 3LCPh states of the dppm aromatic rings. The higher donating ability of the substituents in the pyrazolate ligand (CF3 vs CH3) lowers the energy of the metal-centered excited state, allowing for a significant metal impact on the T1 state. Finally, the Ag(I) complex displays an emission efficiency of approximately 14%, being the highest among known trinuclear silver(I) pyrazolate homometallic derivatives.

One-Step Synthesis of Heterostructured Mo@MoO2 Nanosheets for High-Performance Supercapacitors with Long Cycling Life and High Rate Capability.

Supercapacitors have gained increased attention in recent years due to their significant role in energy storage devices; their impact largely depends on the electrode material. The diversity of energy storage mechanisms means that various electrode materials can provide unique benefits for specific applications, highlighting the growing trend towards nanocomposite electrodes. Typically, these nanocomposite electrodes combine pseudocapacitive materials with carbon-based materials to form heterogeneous structural composites, often requiring complex multi-step preparation processes. This study introduces a straightforward approach to fabricate a non-carbon-based Mo@MoO2 nanosheet composite electrode using a one-step thermal evaporating vapor deposition (TEVD) method. This novel electrode features Mo at the core and MoO2 as the shell and demonstrates exceptional electrochemical performance. Specifically, at a current density of 1 A g-1, it achieves a storage capacity of 205.1 F g-1, maintaining virtually unchanged capacity after 10,000 charge-discharge cycles at 2 A g-1. The outstanding long-cycle stability is ascribed to the vertical two-dimensional geometry, the superior conductivity, and pseudocapacitance of the Mo@MoO2 core-shell nanosheets. These attributes significantly improve the electrode's charge storage capacity, charge transfer speed, and structural integrity during the cycling process. The development of the one-step grown Mo@MoO2 nanosheets offers a promising way for the advancement of high-performance, non-carbon-based supercapacitor nanocomposite electrodes.

Ordered structures formed by nematic topological defects and their transformation with changing the Euler characteristics.

Ordered chain structures from topological defects of opposite charges ("necklaces" of defects) were prepared and their dynamics and cooperative rearrangement were investigated. We studied topological defects in nematic films with change of the Euler characteristic induced by temperature. Topological defects emerged due to competing surface anchoring on the nematic-isotropic and nematic-solid interfaces. Transformation of the structure with a circular chain from topological defects to the structure with a single defect and then to a structure without defects takes place as the nematic geometry changes. The temporal evolution of the number of topological defects at their annihilation in the chains differs from coarsening in two-dimensional (2D) and 3D geometry.

Learners’ algebraic and geometric connections when solving Euclidean geometry riders

In this article, we explored Grade 11 learners’ algebraic and geometric connections when solving Euclidean geometry riders. A qualitative interpretive case study design was followed. Thirty Grade 11 learners from a non-fee-paying secondary school in the Capricorn North district of South Africa were conveniently sampled to participate in this study. Data were collected through learners’ responses to classwork, homework exercises, and task-based interviews. Data were analysed thematically. The findings revealed that to solve Euclidean geometry riders successfully, learners need to establish the feature connections embedded in the given figure or diagram. The ability to make feature connections provides a point of departure in the solution process of a geometric problem.Contribution: Once the feature connection is established, other connections will subsequently emerge. In addition, the reversibility connections become a form of feature connection when solving Euclidean geometry riders. Therefore, we recommend that mathematics teachers emphasise and use mathematical connections in their daily teaching of mathematics.

Fabrication, Preparation, Physicochemical Characterization, and Theoretical Studies of Some Novel Schiff Base Ciprofloxacin Metal Complexes: DNA Interaction and Biomedical Applications

ABSTRACTNovel compounds with pharmacological activity were synthesized from Pd(II), Fe(III), Cr(III), Ni(II), and Cu(II) ions with 1‐cyclopropyl‐6‐fluoro‐4‐(2‐hydroxy‐phenylimino)‐7‐piperazin‐1‐yl‐1,4‐dihydro‐quinoline‐3‐carboxylic acid (CFAP). The ligand's NH and OH groups allowed it to interact with the metals as a neutral tridentate. The investigated novel compounds were described using 1H and 13C NMR spectra, FT‐IR spectrums, TGA and UV–Vis (conductance of molecules), and CHN‐analysis. Additionally, the pH profile of the CFAP complexes indicated great stability, and the complexes' stability constant was discovered in the solution. To extract important properties for CFAP and its complexes, computational research was used, CFAPCu, CFAPCr, CFAPNi, and CFAPFe have octahedral geometry, while CFAPPd has square planar geometry. To investigate the molecular geometry, density functional theory calculations (DFT) were carried out. The molar ratio and continuous fluctuation data confirmed that the (M:L) ratio was (1:1). In vitro tests were conducted to evaluate Schiff base's antimicrobial action ligand and its metal chelates against fungal and bacterial infections. The findings showed that the antimicrobial efficacy is as follows: When CFAPPd is compared to fluconazol and ofloxacin as reference medications, it is the highly inhibitor complex. The novel CFAP ligand and its complexes were investigated for In vitro carcinogenic potential against Hep‐G2, MCF‐7, and HCT‐116 cell lines. As compared to the medication vinblastine, the results once again demonstrated that CFAPPd is the most active agent. Moreover, the complexes demnstrated strong reactivity in capturing free radicals when their antioxidant activity was investigated. Viscosity, spectral investigations, and gel electrophoreses were used to identify the interaction between metal chelates and DNA. Every examined compound is shown to be an enthusiastic DNA binder by viscosity and spectrophotometric titration investigations. The heightened hydrophobic and electrostatic interactions between aromatic rings could be the cause of this. Ultimately, these compounds could be regarded as promising bioactive substances.

Unveiling photon–photon coupling induced transparency and absorption

Abstract This study presents the theoretical foundations of an analogous electromagnetically induced transparency (EIT) and absorption (EIA) which we are referring as coupling induced transparency (CIT) and absorption (CIA) respectively, along with an exploration of the transition between these phenomena. We provide a concise phenomenological description with analytical expressions for transmission spectra and dispersion elucidating how the interplay of coherent and dissipative interactions in a coupled system results in the emergence of level repulsion and attraction, corresponding to CIT and CIA, respectively. This theory comprehensively captures both the phenomena while modelling the microstripline loaded resonators and their couplings systematically. The model is validated through numerical simulations using a hybrid system comprising a split ring resonator (SRR) and electric inductive-capacitive (ELC) resonator in planar geometry. We analyse two cases while keeping ELC parameters constant; one involving a dynamic adjustment of the SRR size with a fixed split gap, and the other entailing a varying gap while maintaining a constant SRR size. Notably, in the first case, the dispersion profile of the transmission signal demonstrates level repulsion, while the second case results in level attraction, effectively showcasing CIT and CIA, respectively. These simulated findings not only align with the theoretical model but also underscore the versatility of our approach. Subsequently, we expand our model to a more general case, demonstrating that a controlled transition from CIT to CIA is achievable by manipulating the dissipation rate of individual modes within the hybrid system, leading to either coherent or dissipative interactions between the modes. The results provide a pathway for designing hybrid systems that can control the group velocity of light, offering potential applications in the fields of optical switching and quantum information technology.

Circling a Regular Pentagon with Straightedge and Compass in Euclidean Geometry

“Circling The Regular Pentagon” is possibly used to describe a new challenge problem that comes from the exact solution for another new challenge problem “Circling the SQUARE”, published in January 2024 in the International Journal of Mathematics Trends and Technology (IJMTT). "Circling The SQUARE" means constructing a circle that has the exact area of the given square”. Therefore, the “Circling the Regular Pentagon” problem challenge has not existed in the mathematics field until it is solved and published nowadays. This paper, then, provides an exact solution to construct a circle that has the same area as a given regular pentagon. The solution does not use the number  and suits the exact constraint of Euclidean Geometry by straightedge & compass.

Structural and computational analysis of a nickel(II) complex with S-benzyldithiocarbazate Schiff base ligand: Synthesis, X-ray crystallography, MEP, HOMO-LUMO, Hirshfeld surface analysis, and molecular docking

A new compound, benzyl N-[1-(thiophen-2-yl)propylidene]hydrazinecarbodithioate (LH), and its nickel(II) complex (NiL2), were synthesized and structurally identified. Single crystal X-ray diffraction analyses determined that both the LH ligand and the NiL2 complex crystallized in the monoclinic system, with space group P21/c and Z = 4. The LH ligand exhibited lattice parameters of a = 9.661(2)Å, b = 8.995(2)Å, c = 18.939(4)Å, β = 102.677(4)°, V = 1605.7(7)Å3, whereas, NiL2 exhibited a = 17.805(7)Å, b = 10.548(4)Å, c = 18.100(7)Å, β = 108.279(6)°, V = 3228(2)Å3. The NiL2 complex was found to possess a slightly distorted square planar geometry with the LH deprotonated ligand acting as a monoanionic bidentate by coordinating through the azomethine nitrogen and thiolate sulfur atoms. It was observed that the HOMO-LUMO energy gap of the NiL2 complex (0.3654 eV) was lower than that of the LH ligand (2.4425 eV), indicating that NiL2 possesses a higher reactivity than the LH ligand. Intermolecular hydrogen bonding interactions were examined by Hirshfeld surface and 2D fingerprint analysis, which showed predominant H…H interactions for both the LH ligand and the NiL2 complex. A molecular docking simulation study with the dopamine D4 receptor (DRD4) and farnesyltransferase (FTase) revealed that NiL2 was able to interact with 13 residues in FTase, as compared to the farnesyl-Cys drug interacted with 11 residues.

Hydrogen embrittlement of Zircaloy-4 fabricated by ultrasonic additive manufacturing

Ultrasonic additive manufacturing (UAM) was successfully applied to the zirconium material system to create a planar geometry. Following fabrication, SS-J3 type tensile specimens of the UAM and wrought Zircaloy-4 with a nominal gage section of 5 × 1.2 × 0.75 mm were machined for hydriding studies and mechanical testing. The SS-J3 type tensile specimens were gas-charged with hydrogen using a custom system that precisely controls hydrogen gas flow rate, hydrogen partial pressure, and temperature. To avoid altering the UAM material, the maximum process temperature was limited to 550 °C. Using different initial hydrogen gas pressures and flow rates, various hydrogen contents (70–1755 wppm) were achieved for Zircaloy-4 specimens. Tensile testing shows that, regardless of hydrogen content, all UAM specimens measured yield strengths in the range of 557 ± 16 MPa and ultimate tensile strengths of 660 ± 4 MPa. However, total elongation clearly decreased as a function of increasing hydrogen content. At the lowest hydrogen content, the total plastic elongation measured 21.5 %, and this value decreased to less than 2 % when the hydrogen content was increased to 1000 wppm. Cross-sectional optical microscopy images revealed that the hydride distributions are randomly oriented. For the low hydrogen content specimen, these randomly distributed hydrides are isolated from each other. A sandwiched structure, consisting of high-density and low-density layers, was also observed for the specimens with hydrogen content between 200 and 400 wppm. As the hydrogen content increases, the hydrides diffuse into the low-density hydride layers to form a network across the whole specimen. Although the tensile properties of UAM and wrought Zircaloy-4 exhibit the same behavior with increasing hydrogen content, the distinctions in grain orientations led to differences in the hydride orientations at low and intermediate hydrogen concentrations.

Mixed ligand complexes of Pt(II) Tertiary phosphines and 1-(benzo[d]thiazol-2-yl)-3-phenylthiourea; Crystal structure and DFT of [Pt(C14H9N3S2)(dppe)].CH2Cl2

This paper describes a series of Pt(II) thiourea complexes (1–4). The complexes were synthesized by a reaction between [PtCl2(diphos)] and the ligand 1-(benzo[d]thiazol-2-yl)-3-phenylthioure in the presence of triethylamine, with all reactants present in equimolar amounts. The produced complexes exhibited square planar geometry, which was validated by FT-IR and 1HNMR . Single crystal XRD study revealed that the coordination geometry was distorted square planar in [Pt(Btzt)(dpppe)]. Hirshfeld surface analysis was used to delve deeper into the intermolecular interactions which showed that the combined contribution of H⋯H and H⋯C contacts in the stabilization of the supramolecular assembly was 81.3%. A void analysis was carried out which showed that there was no large cavity in the supramolecular assembly. Theoretical studies were also conducted on [Pt(Btzt)(dpppe)]. The DFT analysis was carried out using the B3LYP Hybrid method in combination with Def2-SVP. The theoretical findings indicated just 3% differences between the theoretical and experimental Figures. Furthermore, the computed HOMO-LUMO gap was 3.4 eV, which was close to the anatase photocatalyst's band gap. DFT predicted that all complexes are non-magnetic with μB=0. There are high electron donation and back-donation between heteroatoms and transition metals in the complexes.

A Double Legendre Polynomial Order N Benchmark Solution for the 1D Monoenergetic Neutron Transport Equation in Plane Geometry

As more and more numerical and analytical solutions to the linear neutron transport equation become available, verification of the numerical results becomes increasingly important. This presentation concerns the development of another benchmark for the linear neutron transport equation in a benchmark series, each employing a different method of solution. In 1D, there are numerous ways of analytically solving the monoenergetic transport equation, such as the Wiener–Hopf method, based on the analyticity of the solution, the method of singular eigenfunctions, inversion of the Laplace and Fourier transform solution, and analytical discrete ordinates in the limit, which is arguably one of the most straightforward, to name a few. Another potential method is the PN (Legendre polynomial order N) method, where one expands the solution in terms of full-range orthogonal Legendre polynomials, and with orthogonality and series truncation, the moments form an open set of first-order ODEs. Because of the half-range boundary conditions for incoming particles, however, full-range Legendre expansions are inaccurate near material discontinuities. For this reason, a double PN (DPN) expansion in half-range Legendre polynomials is more appropriate, where one separately expands incoming and exiting flux distributions to preserve the discontinuity at material interfaces. Here, we propose and demonstrate a new method of solution for the DPN equations for an isotropically scattering medium. In comparison to a well-established fully analytical response matrix/discrete ordinate solution (RM/DOM) benchmark using an entirely different method of solution for a non-absorbing 1 mfp thick slab with both isotropic and beam sources, the DPN algorithm achieves nearly 8- and 7-place precision, respectively.

Design, Synthesis and Bioactivity Evaluation of Ag(I)-, Au(I)- and Au(III)-Quinoxaline-Wingtip N-Heterocyclic Carbene Complexes Against Antibiotic Resistant Bacterial Pathogens.

Intending to homogenize the biological activities of both quinoxaline and imidazole moieties, the proligand, 1-methyl-3-quinoxaline-imidazolium hexaflurophosphate (1.HPF6), and [Ag(1)2][PF6], (2); [Au(1)2][PF6], (3); and [Au(1)Cl3], (4) NHC complexes were synthesized. All the synthesized compounds were characterized by elemental analysis, NMR, and UV-Vis spectroscopy. Finally, single crystal X-ray structures revealed a linear geometry for complex 2 whereas a square planar geometry for complex 4. The formation of complex 3 was confirmed and supported by its MS spectra. The antibacterial activities of all the synthesized complexes were investigated against gram-positive bacteria and gram-negative bacteria. The Au(III)-NHC complex, 4 showed the highest antibacterial activity with extremely low MIC values against both the bacterial strains (0.24 μg mL-1). Monitoring of zeta potential supports the higher activity of complex 4 compared to 2 and 3. ROS production by complex 4 has also been measured in vitro in the CT26 cancer cell lines, which is directly responsible for targetting and killing the bacterial pathogens. Cell cytotoxicity assay using 293T cell lines has been performed to investigate the biocompatibility nature of complex 4. Also, an excellent hemocompatibility was assigned to it from its hemolytic studies, which provide valuable insights into the design of novel antibacterial agents.

Regional Functionalization Molecular Design Strategy: A Key to Enhancing the Efficiency of Multi-Resonance OLEDs.

Herein, we propose a regional functionalization molecular design strategy that enables independent control of distinct pivotal parameters through different molecule segments. Three novel multiple resonances thermally activated delayed fluorescence (MR-TADF) emitters A-BN, DA-BN, and A-DBN, have been successfully synthesized by integrating highly rigid and three-dimensional adamantane-containing spirofluorene units into the MR framework. These molecules form two distinctive functional parts: part 1 comprises a boron-nitrogen (BN)-MR framework with adjacent benzene and fluorene units forming a central luminescent core characterized by an exceptionally rigid planar geometry, allowing for narrow FWHM values; part 2 includes peripheral mesitylene, benzene, and adamantyl groups, creating a unique three-dimensional "umbrella-like" conformation to mitigate intermolecular interactions and suppress exciton annihilation. The resulting A-BN, DA-BN, and A-DBN exhibit remarkably narrow FWHM values ranging from 18 to 14 nm and near-unity photoluminescence quantum yields. Particularly, OLEDs based on DA-BN and A-DBN demonstrate outstanding efficiencies of 35.0 % and 34.3 %, with FWHM values as low as 22 nm and 25 nm, respectively, effectively accomplishing the integration of high color purity and high device performance.

Regional Functionalization Molecular Design Strategy: A Key to Enhancing the Efficiency of Multi‐Resonance OLEDs

AbstractHerein, we propose a regional functionalization molecular design strategy that enables independent control of distinct pivotal parameters through different molecule segments. Three novel multiple resonances thermally activated delayed fluorescence (MR‐TADF) emitters A‐BN, DA‐BN, and A‐DBN, have been successfully synthesized by integrating highly rigid and three‐dimensional adamantane‐containing spirofluorene units into the MR framework. These molecules form two distinctive functional parts: part 1 comprises a boron‐nitrogen (BN)‐MR framework with adjacent benzene and fluorene units forming a central luminescent core characterized by an exceptionally rigid planar geometry, allowing for narrow FWHM values; part 2 includes peripheral mesitylene, benzene, and adamantyl groups, creating a unique three‐dimensional “umbrella‐like” conformation to mitigate intermolecular interactions and suppress exciton annihilation. The resulting A‐BN, DA‐BN, and A‐DBN exhibit remarkably narrow FWHM values ranging from 18 to 14 nm and near‐unity photoluminescence quantum yields. Particularly, OLEDs based on DA‐BN and A‐DBN demonstrate outstanding efficiencies of 35.0 % and 34.3 %, with FWHM values as low as 22 nm and 25 nm, respectively, effectively accomplishing the integration of high color purity and high device performance.

Proof Without Words: Estimating Generalized Pi

Summary Euclidean geometry is a special case of the p-metric whose unit circle satisfies | x | p + | y | p = 1 , and we may use its area to define a generalized version of pi. We wordlessly deduce some simple bounds on the values of generalized pi from inscribed and exscribed octagons.

Near-threshold fatigue crack propagation in a Ni-based single crystal superalloy affected by crystallographic anisotropy

Fatigue crack propagation in the near-threshold regime was investigated experimentally and numerically in a Ni-based single crystal superalloy, namely, CMSX-4. Fatigue crack propagation tests were conducted at room temperature using four types of specimens with different primary and secondary crystallographic orientations. The results reveal that the crystallographic orientations strongly affect the fatigue crack propagation behavior, including the crack paths, fatigue crack propagation rates, and fatigue threshold. Crystal plasticity finite element analysis was conducted to quantify the slip activity of an octahedral slip system in front of the crack tip, considering the actual 3D geometry of the crack plane and elastic–plastic anisotropy. The fatigue damage parameter, considering the slip activities of the individual octahedral slip systems, provides reasonable explanation for the fatigue crack propagation rates and crack paths in the near-threshold regime, regardless of the crystallographic orientation. Furthermore, these damage parameters are equivalent at the fatigue thresholds for all specimens with different crystallographic orientation.

Discontinuous Galerkin approximations of the heterodimer model for protein–protein interaction

Mathematical models of protein–protein dynamics, such as the heterodimer model, play a crucial role in understanding many physical phenomena, e.g., the progression of some neurodegenerative diseases. This model is a system of two semilinear parabolic partial differential equations describing the evolution and mutual interaction of biological species. This article presents and analyzes a high-order discretization method for the numerical approximation of the heterodimer model capable of handling complex geometries. In particular, the proposed numerical scheme couples a Discontinuous Galerkin method on polygonal/polyhedral grids for space discretization, with a θ-method for time integration. This work presents novelties and progress with respect to the mathematical literature, as stability and a-priori error analysis for the heterodimer model are carried out for the first time. Several numerical tests are performed, which demonstrate the theoretical convergence rates, and show good performances of the method in approximating traveling wave solutions as well as its flexibility in handling complex geometries. Finally, the proposed scheme is tested in a practical test case stemming from neuroscience applications, namely the simulation of the spread of α-synuclein in a realistic test case of Parkinson’s disease in a two-dimensional sagittal brain section geometry reconstructed from medical images.