Synthesis and tautomerism study of 7-substituted pyrazolo[3,4-c]pyridines

A number of 7-substituted pyrazolo[3,4-c]pyridine derivatives have been synthesized in order to investigate the N1-N2 tautomerism within this class of biologically interesting compounds. Tautomeric equilibrium has been studied using NMR ¹³C, ¹⁵N chemical shifts and heteronuclear ¹H-¹⁵N and ¹H-¹³C spin-spin couplings, in conjunction with X-ray crystallography. The N1 tautomer predominates in DMF solution in all the compounds tested.     

Zero point energy of the Pullen–Edmonds Hamiltonian

Here we wish to apply the newly developed Generalized Moments Expansion (GMX) to the well-known potential

The Dyck and the Preiss separation uniformly

We are concerned with two separation theorems about analytic sets by Dyck and Preiss, the former involves the positively-defined subsets of the Cantor space and the latter the Borel-convex subsets of finite dimensional Banach spaces. We show by introducing the corresponding separation trees that both of these results admit a constructive proof. This enables us to give the uniform version of these separation theorems, and to derive as corollaries the results, which are analogous to the fundamental fact “HYP is effectively bi-analytic” provided by the Suslin–Kleene Theorem.     

Turning points in random permutations of two kinds of distinct elements

The mean and variance of the number of turning points in random permutations of two kinds of distinct elements are evaluated. These results are applied to a Wald–Wolfowitz run test.     

On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation

Summary We approximate the solutions of an initial- and boundary-value problem for nonlinear Schrödinger equations (with emphasis on the cubic nonlinearity) by two fully discrete finite element schemes based on the standard Galerkin method in space and two implicit. Crank-Nicolson-type second-order accurate temporal discretizations. For both schemes we study the existence and uniqueness of their solutions and proveL ² error bounds of optimal order of accuracy. For one of the schemes we also analyze one step of Newton's method for solving the nonlinear systems that arise at every time step. We then implement this scheme using an iterative modification of Newton's method that, at each time stept ⁿ , requires solving a number of sparse complex linear systems with a matrix that does not change withn. The effect of this inner iteration is studied theoretically and numerically.The work of these authors was supported by the Institute of Applied and Computational Mathematics of the Research Center of Crete-FORTH and the Science Alliance program of the University of TennesseeThe work of this author was supported by the AFOSR Grant 88-0019     

Students’ use of technological tools for verification purposes in geometry problem solving

Despite its importance in mathematical problem solving, verification receives rather little attention by the students in classrooms, especially at the primary school level. Under the hypotheses that (a) non-standard tasks create a feeling of uncertainty that stimulates the students to proceed to verification processes and (b) computational environments - by providing more available tools compared to the traditional environment - might offer opportunities for more frequent usage of verification techniques, we posed to 5th and 6th graders non-routine problems dealing with area of plane irregular figures. The data collected gave us evidence that computational environments allow the development of verification processes in a wider variety compared to the traditional paper-and-pencil environment and at the same time we had the chance to propose a preliminary categorization of the students’ verification processes under certain conditions.     

Some remarks on a class of rational approximations to the cosine

Some previously introduced methods for the numerical solution of second-order evolution equations based on a class of rational approximations to the cosy, y real, with denominators (1+x ² y ²) ^s are revisited. It is shown that maximal accuracy occurs for each integers1 for a suitable choice of the parameterx. An improved sufficient condition for unconditional stability is obtained. Conditional stability and periodicity of these methods are also studied.Work supported by USARO Grant DAAG 29-78-C-0024.     

An initial- and boundary-value problem for a model equation for propagation of long waves

An initial- and boundary-value problem for a model equation for small-amplitude long waves is shown to be well-posed. The model has the form u_t + u_x + uu_x ? vu_xx ? α²u_xxt = 0, where x? [0, 1] and t ? 0. The solution u = u(x, t) is specified at t = 0 and on the two boundaries x = 0 and x = 1. Unique classical solutions are shown to exist, which depend continuously on variations of the specified data within appropriate function classes.     

Synthesis,Spectroscopy and Crystal Structure Analysis of N1,N3-dicyclohexyl-N1-(all-trans-retinoyl)urea

The title compound, C₃₃H₅₀N₂O₂, is a side product in the reaction of all-trans-retinoic acid (atRA) with N-hydroxysuccinimide, in the presence of the coupling agent N,N′-dicyclohexylcarbodiimide, which produces the ‘active’ ester succinimidyl all-trans-retinoate as the product. It crystallizes in the orthorhombic Pbca space group. The compound was characterized by ¹H-NMR, ¹³C-NMR, ESI–MS and IR spectroscopy and its structure was determined by single-crystal X-ray diffraction. For example in the ¹³C-NMR spectrum, diagnostic peaks are those of the two amide carbonyl C atoms at δ 169.5 and 154.2 ppm, the ten olefinic C atoms of the unsaturated chain of atRA moiety at δ 149.0, 139.3, 137.7, 137.3, 134.9, 130.2, 130.0, 129.4, 128.5 and 121.5 ppm and the two methine C atoms of the N,N′-dicyclohexylurea moiety at δ 57.9 and 49.5 ppm. Detailed analysis of its molecular and supramolecular structure showed that close-packing principles (elongated shape/large hydrophobic region of the molecule) together with chemical factors (N–H?O and C–H?O intermolecular interactions) direct the 3D self-assembly process in the crystalline state. Hirshfeld surface analysis was employed, a powerful approach to quickly and easily gain insight into molecular environments in the crystalline state.

Graphical Abstract

The synthesis and X-ray structure of 1-((2E, 4E, 6E, 8E)-3,7-dimethyl-9-(2,6,6-trimethylcyclohex-1-enyl)nona-2,4,6,8-tetraenoyl)-1,3-dicyclohexylurea, a side product in the synthesis of succinimidyl all-trans-retinoate, is reported; Hirshfeld surface analysis was employed to identify intermolecular interactions.