Grignard reactions in imidazolium ionic liquids

A new, base-stable, imidazolium room-temperature ionic liquid (RTIL) has been prepared and applied to the addition of Grignard reagents to carbonyl compounds. These reactions occur readily at ambient temperature to afford the alcohol products in good to excellent yield. The RTIL can be recycled and reused numerous times without any difficulty.     

The weakly coordinating trichloromethanesulfonate anion: NQR comparison of its coordinating abilities via oxygen with those of the chloroacetate ions

Ionic and covalent derivatives of the chlorine analogue of the nonbasic, weakly coordinating triflate ion, Cl3CSO3(-) or "trichlate" ion, have been prepared and compared with the corresponding more strongly coordinated chloroacetates, Cl(x)CH(3-x)CO(2)M (x = 1-3), using 35Cl NQR (nuclear quadrupole resonance) spectroscopy. The (35)Cl NQR frequencies of all types of derivatives are sensitive to the nature of the metal ion or Lewis acid and are most sensitive in the case of monochloroacetates. In covalent (including zirconocene) derivatives, the average NQR frequencies fall as the Pauling electronegativity of M falls. The results for ionic derivatives contrast with previous results for ionic hexachlorometalates: the average (35)Cl NQR frequencies drop sharply as the ionic radius of the group 1 cation increases. Ab initio Gaussian 98 computations at the B3LYP/6-311++G(3df,3pd) level on isolated XCH2CO2M (M = Li, Na, K; X = F, Cl) molecules duplicate this trend, showing increasing polarization of the C-Cl bond and smaller electric field gradients for larger group 1 ions; the relevance of this to the solid state polymerization of chloroacetates (Herzberg, O.; Epple, M. Eur. J. Inorg. Chem. 2001, 1395-1406) is discussed. We have prepared the dihydrate and monohydrate of trichlic acid, Cl3CSO3H. Although trichlates have the highest average NQR frequencies of any of these salts, the NQR frequencies of trichlic acid dihydrate are anomalously lower than those of trichloroacetic acid, which suggests that it is a strong acid, ionized in the solid state to H5O2+ and Cl3CSO3(-) ions.     

Weakly coordinating anions: crystallographic and NQR studies of halogen-metal bonding in silver, thallium, sodium, and potassium halomethanesulfonates

35Cl, (79,81)Br, and (127)I NQR (nuclear quadrupole resonance) spectroscopy in conjunction with X-ray crystallography is potentially one of the best ways of characterizing secondary bonding of metal cations such as Ag(+) to halogen donor atoms on the surfaces of very weakly coordinating anions. We have determined the X-ray crystal structure of Ag(O(3)SCH(2)Cl) (a = 13.241(3) A; b = 7.544(2) A; c = 4.925(2) A; orthorhombic; space group Pnma; Z = 4) and compared it with the known structure of Ag(O(3)SCH(2)Br) (Charbonnier, F.; Faure, R.; Loiseleur, H. Acta Crystallogr., Sect. B 1978, 34, 3598-3601). The halogen atom in each is apical (three-coordinate), being weakly coordinated to two silver ions. (127)I NQR studies on Ag(O(3)SCH(2)I) show the expected NQR consequences of three-coordination of iodine: substantially reduced NQR frequencies nu(1) and nu(2) and a fairly small NQR asymmetry parameter eta. The reduction of the halogen NQR frequency of the coordinating halogen atom in Ag(O(3)SCH(2)X) becomes more substantial in the series X = Cl < Br < I, indicating that the coordination to Ag(+) strengthens in this series, as expected from hard-soft acid-base principles. The numbers of electrons donated by the organic iodine atom to Ag(+) have been estimated; these indicate that the bonding to the cation is weak but not insignificant. We have not found any evidence for the bonding of these organohalogen atoms to another soft-acid metal ion, thallium. A scheme for recycling of thallium halide wastes is included.     

Efficient representations of Green’s functions for some elliptic equations with piecewise-constant coefficients

Convenient for immediate computer implementation equivalents of Green’s functions are obtained for boundary-contact value problems posed for two-dimensional Laplace and Klein-Gordon equations on some regions filled in with piecewise homogeneous isotropic conductive materials. Dirichlet, Neumann and Robin conditions are allowed on the outer boundary of a simply-connected region, while conditions of ideal contact are assumed on interface lines. The objective in this study is to widen the range of effective applicability for the Green’s function version of the boundary integral equation method making the latter usable for equations with piecewise-constant coefficients.     

Extinction and blow-up phenomena in a non-linear gender structured population model

In this article we consider a gender structured model in population dynamics. We assume that the fertility rate depends upon the weighted population of males instead of total population of males. The proportion of males in the population is determined by fixed environmental or social conditions. Here we prove an existence and uniqueness result for a non-trivial steady state. If the initial age distribution is uniformly below the non-trivial steady state then we show that the total population goes extinct in infinite time. On the other hand, if we take the initial age distribution to be uniformly above the steady state then the total population blows up exponentially with time.     

Thickness and outerthickness for embedded graphs

We prove new upper bounds for the thickness and outerthickness of a graph in terms of its orientable and nonorientable genus by applying the method of deleting spanning disks of embeddings to approximate the thickness and outerthickness. We also show that every non-planar toroidal graph can be edge partitioned into a planar graph and an outerplanar graph. This implies that the outerthickness of the torus (the maximum outerthickness of all toroidal graphs) is 3. Finally, we show that all graphs embeddable in the double torus have thickness at most 3 and outerthickness at most 5.     

Efficient high-order compact exponential time differencing method for space-fractional reaction-diffusion systems with nonhomogeneous boundary conditions

Numerical Algorithms - This paper introduces an efficient unconditionally stable fourth-order method for solving nonlinear space-fractional reaction-diffusion systems with nonhomogeneous Dirichlet...     

An Emergent Framework: Views of Mathematical Processes

The findings reported in this paper were generated from a case study of teacher leaders at a state‐level mathematics conference. Investigation focused on how participants viewed the mathematical processes of communication, connections, representations, problem solving, and reasoning and proof. Purposeful sampling was employed to select nine participants who were then interviewed and observed as they presented a session at the conference. Participants' statements revealed differences in their views of mathematical processes. The analysis led to an emergent framework for views of mathematical processes that includes three levels: participatory, experiential, and sense‐making. Implications are shared for mathematics methods instructors, professional learning, and research. Discussion also relates the framework to the Standards for Mathematical Practice.