Love stories in a differential equations classroom

We believe that developing cultural competencies can help students learn mathematics and conversely that learning mathematical content can help students learn about themselves and others. Using frameworks introduced by Rendón (2009) and Gutiérrez (2018), we present a five-part bundle of activities for undergraduate differential equations course instructors, including one pre-activity reflection assignment, three modeling activities, and a final project. The Pre-Activity Assignment engages students to draw on their own personal and/or cultural experiences with the concept of love. Three activities focus on developing revision skills in mathematical modeling and practicing methods of solving systems of ordinary differential equations (ODEs). In these activities, students would collaborate to construct a relationship model consisting of a system of first-order linear ODEs and solve different model iterations with the characteristic polynomial, matrix form, or Laplace transform method. The final project connects the reflections in Pre-Activity Assignment with the skills developed in the three activities by inviting students to create a relationship scenario, model, revise, solve it, and present the conclusions. By engaging in this set of assignments, students connect personal and cultural experiences with the concept of love and perceive themselves and their peers in the curriculum, fostering a sense of belonging and relevance.</p

Coarse-Grained Molecular Dynamics Simulations of the Phase Behavior of the 4-Cyano-4′-pentylbiphenyl Liquid Crystal System

In this paper, with the aim to establish a rational coarse-grained (CG) model for the 4-cyano-4′-pentylbiphenyl (5CB) molecule, we construct three possible CG models (5P, 6P, and 7P) and then determine the bonded and nonbonded interaction parameters separately. For the intramolecular bonded interactions, the bond and angle distributions of the 5CB bulk phase are used as the target properties. For the nonbonded interactions between CG particles, we combine the structure-based and thermodynamic quantities-based methods for the parametrization of CG interaction potentials and attempt to use several fragment molecular systems to derive the CG nonbonded interaction parameters in order to maintain the transferability of our CG models to some extent. Finally, we fix the optimal nonbonded LJ parameters between CG bead pairs such that the results from CG simulations not only correctly reproduce the experimental density and the nematic LC state at 300 K and 1 atm but also reasonably approximate the local structural properties calculated from the underlying atomistic model. Through comparison of the resulting CG data with target properties, the 6P model is found to be the best one among the three, and then we use this model to investigate the phase behavior and dynamic properties. Our results show that the phase transition temperature from nematic to isotropic phase and the diffusion coefficients are reproduced very well, demonstrating the rationality of the 6P model. Our coarse-grained process should have implications for constructing CG models for nCB series or molecules with similar architectures

Developments in compartmentalized bimetallic transition metal ethylene polymerization catalysts

Recent progress concerning the application of compartmentalized bimetallic complexes as homogeneous catalysts in ethylene polymerization is reviewed with particular regard to metal-metal combinations based on either early- (Ti, Zr, Hf and V) or late-transition metals (Fe, Co and Ni). The effect of positioning two polymerization-active metal centers in close proximity on catalytic activity, molecular weight, molecular weight distribution and levels of branching are thoroughly documented. Compartmental ligands comprising binding domains consisting of phenoxyimines, ansa-bridged cyclopentadienyl-amides, α-diimines and iminopyridines are described as is their capacity to serve as compatible binucleating supports for homobimetallic and also for the less investigated heterobimetallic counterparts. By comparison with their mononuclear analogues, any synergic properties exhibited by these binuclear catalysts represents an underlying theme to be developed where possible throughout this review

Exploring ortho‐(4,4′‐dimethoxybenzhydryl) substitution in iron ethylene polymerization catalysts: Co‐catalyst effects, thermal stability, and polymer molecular weight variations

Six different types of 4,4′-dimethoxybenzhydryl-substituted bis(arylimino)pyridine-iron(II) chloride complex—[2-{{2,6-((p-MeOPh) CH) -4-MeC H }N=CMe}-6-(ArN=CMe)C H N]FeCl (Ar = 2,6-Me C H (Fe1), 2,6-Et C H (Fe2), 2,6- Pr C H (Fe3), 2,4,6-Me C H (Fe4), 2,6-Et -4-MeC H (Fe5), 2,6-((p-MeOPh) CH) -4-MeC H (Fe6)—are reported. The molecular structures of Fe2 and Fe3 emphasize the unsymmetrical nature of the N,N,N-chelating ligand and the steric protection exerted by the ortho-(4,4′-dimethyoxybenzhydryl) groups. A range in catalytic activities for ethylene polymerization was observed when Fe1–Fe6 were treated with either methylaluminoxane (MAO) or modified methylaluminoxane (MMAO). In particular, Fe1/MAO bearing the least bulky N-2,6-dimethylphenyl group exhibited a very high activity of 1.11 × 10 g (PE) mol (Fe) h at 70°C/10 atm over a 30-min run time that remained high even after 60 min, an observation highlighting its appreciable catalyst lifetime and thermal stability. In addition, the polyethylene formed by this catalyst class displayed desirable characteristics such as high linearity and high molecular weight (M range: 5.17–7.62 × 10 g mol ). Indeed, the molecular weight of the polymer produced using Fe1 (with MAO or MMAO) exceeds that obtained using related benzhydryl-containing bis(imino)pyridine-iron catalysts. As a general feature, activation of Fe1–Fe6 with MMAO led to lower activity and lower molecular weight polyethylene, especially for runs performed at 1 atm ethylene pressure. 2 2 6 2 5 3 2 2 6 3 2 6 3 2 6 3 3 6 2 2 6 2 2 2 6 2 w i 7 −1 −1 5 −

Construction of a Prognosis Model of the Pyroptosis-Related Gene in Multiple Myeloma and Screening of Core Genes

Pyroptosis is an important factor affecting the proliferation, invasion, and metastasis of tumor cells. However, in multiple myeloma (MM), there are few studies on whether the occurrence of pyroptosis is related to the occurrence and prognosis of the disease. Based on the Gene Expression Omnibus and Cancer Genome Atlas database search dataset, this study identified pyroptosis-related genes with a specific prognosis, constructed and verified the prediction model by stepwise Cox regression analysis and time receiver operating characteristic curve analysis, and predicted specific functions by single-sample gene set enrichment analysis and the Kyoto Encyclopedia of Genes and Genomes. Dataset analysis identified key genes, which were used to construct a risk scoring system for the prognosis of MM. The entire test set and external verification set verified the results. The expression levels of related genes in the clinical samples were detected using fluorescence quantitative PCR. A prognostic gene model based on six pyroptosis-related genes (CYCS, NLRP9, AIM2, NOD2, CHMP3, and GSDME) was constructed. The model has an excellent prognostic ability and can be popularized in the external validation set. The predictive prognostic nomogram integrating clinical information can effectively evaluate the risk score of each patient and predict their survival. After sample validation, our study found three potential key pyroptosis-related genes in multiple myeloma. GSDME, NOD2, and CHMP3 were significantly different between MM and healthy subjects, suggesting that they are pyroptosis-related protective genes. This study shows that the key pyroptosis-related gene in the model can be used as a marker for predicting the prognosis of myeloma, which may provide a basis for clinical individualized stratification therapy

CF3O-Functionalized Bis(arylimino)pyridine-Cobalt Ethylene Polymerization Catalysts: Harnessing Solvent Effects on Performance and Polymer Properties

Enhancing the properties of polymeric materials through the fine control of molecular weight and dispersity provides key objectives to be considered in the design of polymerization catalysts. Herein, solution polymerization of ethylene was studied using bis(arylimino)pyridine-cobaltous chloride precatalysts, [2-[CMeN{2,6-{(C6H5)2CH}2-4-(F3CO)C6H2}]-6-(CMeNAr)C5H3N]CoCl2 [Ar = 2,6-Me2C6H3 (Co1), 2,6-Et2C6H3 (Co2), 2,6-i-Pr2C6H3 (Co3), 2,4,6-Me3C6H2 (Co4), or 2,6-Et2-4-MeC6H2 (Co5)], each incorporating one N-aryl group appended with both o-benzhydryl and p-trifluoromethoxy groups. In the presence of MAO or MMAO, all complexes displayed very high activities [≤11.2 × 106 g (PE) mol-1 (Co) h-1 for Co1/MAO] as ethylene polymerization catalysts generating high-molecular weight polyethylene (≤5.05 × 105 g mol-1) with narrow dispersity (Mw/Mn ≥ 1.76). Notably, higher activity was achieved in hexane than in toluene with the added economic benefit that less aluminoxane activator was required [e.g., 650:1 Al:Co (hexane) vs 1750:1 Al:Co (toluene)]. All polyethylenes were of high linearity as evidenced by 1H and 13C nuclear magnetic resonance spectroscopy and differential scanning calorimetry. In addition to the in-depth polymerization studies, the synthetic details for Co1-Co5 and their precursor bis(arylimino)pyridines are reported along with associated characterization data, including the X-ray structures for Co2 and Co5

An air and moisture tolerant iminotrihydroquinoline-ruthenium(ii) catalyst for the transfer hydrogenation of ketones.

Reaction of 8-amino-5,6,7,8-tetrahydroquinoline with RuCl2(PPh3)3 at room temperature affords the ruthenium(ii) chelate (8-NH2-C9H10N)RuCl2(PPh3)2 (E), in which the two triphenylphosphine ligands are disposed mutually cis. By contrast, when the reaction is performed at reflux ligand oxidation/dehydrogenation occurs along with cis-trans reorganization of the triphenylphosphines to form the 8-imino-5,6,7-trihydroquinoline-ruthenium(ii) complex, (8-NH-C9H9N)RuCl2(PPh3)2 (F). Complex F can also be obtained in higher yield by heating a solution of E alone to reflux. Comparison of their molecular structures highlights the superior binding properties of the bidentate imine ligand in F over its amine-containing counterpart in E. Both complexes are highly effective in the transfer hydrogenation of a wide range of alkyl-, aryl- and cycloalkyl-containing ketones affording their corresponding secondary alcohols with loadings of as low as 0.1 mol%. Significantly, F can deliver excellent conversions even in bench quality 2-propanol in reaction vessels open to the air, whereas the catalytic efficiency of E is diminished by the presence of air but only operates efficiently under inert conditions

Fusing Carbocycles of Inequivalent Ring Size to a Bis(imino)pyridine-Iron Ethylene Polymerization Catalyst: Distinctive Effects on Activity, PE Molecular Weight, and Dispersity

The 4,6-bis(arylimino)-1,2,3,7,8,9,10-heptahydrocyclohepta[b]quinoline-iron(II) chlorides (aryl = 2,6-Me2C6H3Fe1; 2,6-Et2C6H3Fe2; 2,6-i-Pr2C6H3Fe3; 2,4,6-Me3C6H2Fe4; and 2,6-Et2-4-Me2C6H2Fe5) have been prepared in good yield by a straightforward one-pot reaction of 2,3,7,8,9,10-hexahydro-1H-cyclohepta[b]quinoline-4,6-dione, FeCl2·4H2O, and the appropriate aniline in acetic acid. All ferrous complexes have been characterized by elemental analysis and FT-IR spectroscopy. In addition, the structure of Fe3 has been determined by single crystal X-ray diffraction, which showed the iron center to adopt a distorted square pyramidal geometry with the saturated sections of the fused six- and seven-membered carbocycles to be cis-configured. In combination with either MAO or MMAO, Fe1–Fe5 exhibited exceptionally high activities for ethylene polymerization (up to 15.86×106 gPE mol−1 Fe h−1 at 40°C (MMAO) and 9.60×106 gPE mol−1 Fe h−1 at 60°C (MAO)) and produced highly linear polyethylene (HLPE, Tm≥128°C) with a wide range in molecular weights; in general, the MMAO-promoted polymerizations were more active. Irrespective of the cocatalyst employed, the 2,6-Me2-substituted Fe1 and Fe4 proved the most active while the more sterically hindered 2,6-i-Pr2Fe3 the least but afforded the highest molecular weight polyethylene (Mw: 65.6–72.6 kg mol-1). Multinuclear NMR spectroscopic analysis of the polymer formed using Fe4/MMAO at 40°C showed a preference for fully saturated chain ends with a broad bimodal distribution a feature of the GPC trace (Mw/Mn=13.4). By contrast, using Fe4/MAO at 60°C a vinyl-terminated polymer of lower molecular weight (Mw=14.2 kg mol−1) was identified that exhibited a unimodal distribution (Mw/Mn=3.8). Moreover, the amount of aluminoxane cocatalyst employed, temperature, and run time were also found to be influential on the modality of the polymer

Recent progress in the application of group 1, 2 & 13 metal complexes as catalysts for the ring opening polymerization of cyclic esters

High performance biodegradable polymers (e.g., aliphatic polyesters) that can display properties that rival polyolefins are seen as future high demand materials for a variety of applications. Importantly, these types of polymers can be accessed by the ring opening polymerization (ROP) of monomers that can, in some cases, be derived from cheap biorenewable resources highlighting the sustainability of the process. Carefully designed metal complexes that can act as catalysts for such transformations have emerged as useful tools to achieve this goal. This review is concerned with recent progress in the use of well-defined metal complexes based on group 1, 2 and 13 metals to mediate the formation of aliphatic polyesters with a focus on the role played by the auxiliary ligand on influencing catalytic efficiency, controllability, molecular weight as well as stereoselectivity. More specifically, we report on developments in the design, synthesis and structure of such main group metal species supported by various multidentate ligands including bidentate, tridentate and tetradentate families bearing nitrogen, oxygen, sulfur, selenium or phosphorus donor atoms and their catalytic applications in the ROP of cyclic esters. In addition, the fundamental coordination chemistry of the metal complexes is discussed alongside variations in catalytic performance

Cobalt Catalysts Bearing ortho-(4,4 '-Dichlorobenzhydryl) Substituents and Their Use in Generating Narrowly Dispersed Polyethylene of High Linearity

Six well-defined examples of ortho-(4,4′-dichlorobenzhydryl)-substituted 2,6-diaryliminopyridine-cobalt dichloride complex, Co1−Co6, have been prepared in reasonable yield by treatment of the corresponding free ligand (L1−L6) with cobalt(II) chloride hexahydrate. The molecular structures of Co2 and Co3 highlight their pseudo-square pyramidal geometry, the nonsymmetrical properties of the N,N,N′-chelating ligand and the steric shielding conferred by the ortho-(4,4′-dichlorobenzhydryl) groups. On treatment with the aluminoxanes, MAO or MMAO, all cobalt complexes exhibited good to high productivities for ethylene polymerization (up to 8.96×106 g (PE) mol−1 (Co) h−1 at 60 °C) affording linear polyethylene with molecular weights in the range 28.8–250.2 kg mol−1. The steric properties of the ortho-substituents were found to exhibit significant effects on the activity and molecular weight of the resulting polymers. For example, Co1/MAO proved the most active and thermally stable, while the more sterically encumbered Co3 in combination with either MAO or MMAO afforded the highest molecular weight polymer. All polymerizations were well-controlled as is evidenced by the narrow dispersities displayed by the polymers, a feature that is characteristic of single-site behavior. By comparison with their ortho-benzhydryl-substituted cobalt counterparts, the effect of the para-chloride substitution was to increase thermal stability, raise activity and depress polymer molecular weight